Theorem dmmpog 6188

Description: Domain of an operation given by the maps-to notation, closed form of dmmpo 6184. Caution: This theorem is only valid in the very special case where the value of the mapping is a constant! (Contributed by Alexander van der Vekens, 1-Jun-2017.) (Proof shortened by AV, 10-Feb-2019.)

Hypothesis

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

Assertion

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

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

Allowed substitution hints:   F( x, y)

Proof of Theorem dmmpog

Step	Hyp	Ref	Expression
1		simpl 108	. . 3 |- ( ( C e. V /\ ( x e. A /\ y e. B ) ) -> C e. V )
2	1	ralrimivva 2552	. 2 |- ( C e. V -> A. x e. A A. y e. B C e. V )
3		dmmpog.f	. . 3 |- F = ( x e. A , y e. B |-> C )
4	3	dmmpoga 6187	. 2 |- ( A. x e. A A. y e. B C e. V -> dom F = ( A X. B ) )
5	2, 4	syl 14	1 |- ( C e. V -> dom F = ( A X. B ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   A.wral 2448   X. cxp 4609   dom cdm 4611   e. cmpo 5855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120

This theorem is referenced by: (None)