Theorem dmmpog 6418

Description: Domain of an operation given by the maps-to notation, closed form of dmmpo 6413. 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 2626	. 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 6417	. 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 1398   e. wcel 2205   A.wral 2522   X. cxp 4752   dom cdm 4754   e. cmpo 6060

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348

This theorem is referenced by: (None)