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Theorem dmmpo 8069
Description: Domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
Hypotheses
Ref Expression
fmpo.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
fnmpoi.2 𝐶 ∈ V
Assertion
Ref Expression
dmmpo 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 dmmpo
StepHypRef Expression
1 fmpo.1 . . 3 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
2 fnmpoi.2 . . 3 𝐶 ∈ V
31, 2fnmpoi 8068 . 2 𝐹 Fn (𝐴 × 𝐵)
43fndmi 6652 1 dom 𝐹 = (𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wcel 2099  Vcvv 3469   × cxp 5670  dom cdm 5672  cmpo 7416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988
This theorem is referenced by:  1div0  11895  swrd00  14618  swrd0  14632  pfx00  14648  pfx0  14649  repsundef  14745  cshnz  14766  imasvscafn  17510  imasvscaval  17511  iscnp2  23130  xkococnlem  23550  ucnima  24173  ucnprima  24174  tngtopn  24554  1div0apr  30265  smatlem  33334  elunirnmbfm  33807  rrxsphere  47744
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