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Theorem dmmpt2ga 7131
 Description: Domain of an operation given by the "maps to" notation, closed form of dmmpt2 7129. (Contributed by Alexander van der Vekens, 10-Feb-2019.)
Hypothesis
Ref Expression
dmmpt2g.f 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
dmmpt2ga (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → dom 𝐹 = (𝐴 × 𝐵))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑉,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem dmmpt2ga
StepHypRef Expression
1 dmmpt2g.f . . 3 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
21fnmpt2 7127 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑉𝐹 Fn (𝐴 × 𝐵))
3 fndm 5904 . 2 (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
42, 3syl 17 1 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → dom 𝐹 = (𝐴 × 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896   × cxp 5036  dom cdm 5038   Fn wfn 5799   ↦ cmpt2 6551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060 This theorem is referenced by:  dmmpt2g  7132  mpt2curryd  7282  mamudm  20013  mavmuldm  20175  mavmul0g  20178
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