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Theorem dmmpt2 7390
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 7389 . 2 𝐹 Fn (𝐴 × 𝐵)
4 fndm 6130 . 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 1631  wcel 2145  Vcvv 3351   × cxp 5247  dom cdm 5249   Fn wfn 6026  cmpt2 6795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316
This theorem is referenced by:  1div0  10888  swrd00  13626  swrd0  13643  repsundef  13727  cshnz  13747  imasvscafn  16405  imasvscaval  16406  iscnp2  21264  xkococnlem  21683  ucnima  22305  ucnprima  22306  tngtopn  22674  1div0apr  27666  smatlem  30203  elunirnmbfm  30655  pfx00  41912  pfx0  41913
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