Theorem dmmpog 6234

Description: Domain of an operation given by the maps-to notation, closed form of dmmpo 6230. 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 2572	. 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 6233	. 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 1364   e. wcel 2160   A.wral 2468   X. cxp 4642   dom cdm 4644   e. cmpo 5898

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166

This theorem is referenced by: (None)