Theorem dmmpoga 6444

Description: Domain of an operation given by the maps-to notation, closed form of dmmpo 6440. (Contributed by Alexander van der Vekens, 10-Feb-2019.)

Hypothesis

Ref	Expression
dmmpog.f	|- F = ( x e. A , y e. B |-> C )

Assertion

Ref	Expression
dmmpoga	|- ( A. x e. A A. y e. B C e. V -> dom F = ( A X. B ) )

Distinct variable groups:   x, A, y   x, B, y   x, V, y

Allowed substitution hints:   C( x, y)   F( x, y)

Proof of Theorem dmmpoga

Step	Hyp	Ref	Expression
1		dmmpog.f	. . 3 |- F = ( x e. A , y e. B |-> C )
2	1	fnmpo 6438	. 2 |- ( A. x e. A A. y e. B C e. V -> F Fn ( A X. B ) )
3		fndm 5480	. 2 |- ( F Fn ( A X. B ) -> dom F = ( A X. B ) )
4	2, 3	syl 14	1 |- ( A. x e. A A. y e. B C e. V -> dom F = ( A X. B ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   A.wral 2528   X. cxp 4772   dom cdm 4774   Fn wfn 5372   e. cmpo 6087

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375

This theorem is used by:  dmmpog  6445  mpoexw  6449