Theorem dmmpog 6210

Description: Domain of an operation given by the maps-to notation, closed form of dmmpo 6206. 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 109	. . 3 |- ( ( C e. V /\ ( x e. A /\ y e. B ) ) -> C e. V )
2	1	ralrimivva 2559	. 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 6209	. 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 104   = wceq 1353   e. wcel 2148   A.wral 2455   X. cxp 4625   dom cdm 4627   e. cmpo 5877

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142

This theorem is referenced by: (None)