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Theorem dmmpo 7911
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 7910 . 2 𝐹 Fn (𝐴 × 𝐵)
43fndmi 6537 1 dom 𝐹 = (𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wcel 2106  Vcvv 3432   × cxp 5587  dom cdm 5589  cmpo 7277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832
This theorem is referenced by:  1div0  11634  swrd00  14357  swrd0  14371  pfx00  14387  pfx0  14388  repsundef  14484  cshnz  14505  imasvscafn  17248  imasvscaval  17249  iscnp2  22390  xkococnlem  22810  ucnima  23433  ucnprima  23434  tngtopn  23814  1div0apr  28832  smatlem  31747  elunirnmbfm  32220  rrxsphere  46094
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