Where was the city of Tenochtitlan built?

Before Tenochtitlan was founded, the Aztecs were a nomadic people whose religion stated that whenever they saw a golden eagle, perched atop a cactus and devouring a snake, they had to settle and build a 'great city'. They found such sign in a swampy, small island in Lake Texcoco. This prophecy was immortalized in the Mexican coat of arms.

Now, not deterred by the unfavourable terrain, they set about building their city, using the chinampa system (misnamed as "floating gardens") for agriculture and to dry and expand the island. Basically, such system requires building levees and then filling the reclaimed land with soil. Such system was also implemented - faster and with better technology, of course - by the Dutch people to reclaim land from the sea.
The wandering "Aztec" tribe saw a vision of an eagle on a cactus, and this was foretold to them that this sign would signal the place where they should settle and build their city on a "promised land".