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Theorem dmmpt2 7192
 Description: Domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)
Hypotheses
Ref Expression
fmpt2.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
fnmpt2i.2 𝐶 ∈ V
Assertion
Ref Expression
dmmpt2 dom 𝐹 = (𝐴 × 𝐵)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem dmmpt2
StepHypRef Expression
1 fmpt2.1 . . 3 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
2 fnmpt2i.2 . . 3 𝐶 ∈ V
31, 2fnmpt2i 7191 . 2 𝐹 Fn (𝐴 × 𝐵)
4 fndm 5953 . 2 (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
53, 4ax-mp 5 1 dom 𝐹 = (𝐴 × 𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1480   ∈ wcel 1987  Vcvv 3189   × cxp 5077  dom cdm 5079   Fn wfn 5847   ↦ cmpt2 6612 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121 This theorem is referenced by:  1div0  10638  swrd00  13364  swrd0  13380  repsundef  13463  cshnz  13483  imasvscafn  16129  imasvscaval  16130  iscnp2  20966  xkococnlem  21385  ucnima  22008  ucnprima  22009  tngtopn  22377  1div0apr  27195  smatlem  29669  elunirnmbfm  30120  pfx00  40709  pfx0  40710
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