Theorem ovmpo 5988

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 5987	. 2 |- ( ( A e. C /\ B e. D /\ S e. _V ) -> ( A F B ) = S )
6	1, 5	mp3an3 1321	1 |- ( ( A e. C /\ B e. D ) -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   _Vcvv 2730  (class class class)co 5853   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-in1 609  ax-in2 610  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-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  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-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  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-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858

This theorem is referenced by:  ixxval  9853  fzval  9967