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Theorem dmmpo 8096
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 8095 . 2 𝐹 Fn (𝐴 × 𝐵)
43fndmi 6672 1 dom 𝐹 = (𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  Vcvv 3480   × cxp 5683  dom cdm 5685  cmpo 7433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015
This theorem is referenced by:  1div0  11922  1div0OLD  11923  swrd00  14682  swrd0  14696  pfx00  14712  pfx0  14713  repsundef  14809  cshnz  14830  imasvscafn  17582  imasvscaval  17583  iscnp2  23247  xkococnlem  23667  ucnima  24290  ucnprima  24291  tngtopn  24671  1div0apr  30487  smatlem  33796  elunirnmbfm  34253  rrxsphere  48669
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