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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 3470   × 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 2167  ax-ext 2699  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 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  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  11897  swrd00  14620  swrd0  14634  pfx00  14650  pfx0  14651  repsundef  14747  cshnz  14768  imasvscafn  17512  imasvscaval  17513  iscnp2  23136  xkococnlem  23556  ucnima  24179  ucnprima  24180  tngtopn  24560  1div0apr  30271  smatlem  33392  elunirnmbfm  33865  rrxsphere  47815
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