Theorem ovmpo 6054

Description: Value of an operation given by a maps-to rule. Special case. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Hypotheses

Ref	Expression
ovmpog.1	|- ( x = A -> R = G )
ovmpog.2	|- ( y = B -> G = S )
ovmpog.3	|- F = ( x e. C , y e. D |-> R )
ovmpo.4	|- S e. _V

Assertion

Ref	Expression
ovmpo	|- ( ( A e. C /\ B e. D ) -> ( A F B ) = S )

Distinct variable groups:   x, y, A   x, B, y   x, C, y   x, D, y   x, S, y

Allowed substitution hints:   R( x, y)   F( x, y)   G( x, y)

Proof of Theorem ovmpo

Step	Hyp	Ref	Expression
1		ovmpo.4	. 2 |- S e. _V
2		ovmpog.1	. . 3 |- ( x = A -> R = G )
3		ovmpog.2	. . 3 |- ( y = B -> G = S )
4		ovmpog.3	. . 3 |- F = ( x e. C , y e. D |-> R )
5	2, 3, 4	ovmpog 6053	. 2 |- ( ( A e. C /\ B e. D /\ S e. _V ) -> ( A F B ) = S )
6	1, 5	mp3an3 1337	1 |- ( ( A e. C /\ B e. D ) -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   _Vcvv 2760  (class class class)co 5918   e. cmpo 5920

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-in1 615  ax-in2 616  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-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923

This theorem is referenced by:  ixxval  9962  fzval  10076