Theorem dmmpoga 6372

Description: Domain of an operation given by the maps-to notation, closed form of dmmpo 6368. (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 6366	. 2 |- ( A. x e. A A. y e. B C e. V -> F Fn ( A X. B ) )
3		fndm 5429	. 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 ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   A.wral 2510   X. cxp 4723   dom cdm 4725   Fn wfn 5321   e. cmpo 6019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303

This theorem is referenced by:  dmmpog  6373  mpoexw  6377