Theorem ovmpod 5969

Description: Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 7-Dec-2014.)

Hypotheses

Ref	Expression
ovmpod.1	|- ( ph -> F = ( x e. C , y e. D |-> R ) )
ovmpod.2	|- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S )
ovmpod.3	|- ( ph -> A e. C )
ovmpod.4	|- ( ph -> B e. D )
ovmpod.5	|- ( ph -> S e. X )

Assertion

Ref	Expression
ovmpod	|- ( ph -> ( A F B ) = S )

Distinct variable groups:   x, y, A   x, B, y   x, S, y   ph, x, y

Allowed substitution hints:   C( x, y)   D( x, y)   R( x, y)   F( x, y)   X( x, y)

Proof of Theorem ovmpod

Step	Hyp	Ref	Expression
1		ovmpod.1	. 2 |- ( ph -> F = ( x e. C , y e. D |-> R ) )
2		ovmpod.2	. 2 |- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S )
3		eqidd 2166	. 2 |- ( ( ph /\ x = A ) -> D = D )
4		ovmpod.3	. 2 |- ( ph -> A e. C )
5		ovmpod.4	. 2 |- ( ph -> B e. D )
6		ovmpod.5	. 2 |- ( ph -> S e. X )
7	1, 2, 3, 4, 5, 6	ovmpodx 5968	1 |- ( ph -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136  (class class class)co 5842   e. cmpo 5844

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847

This theorem is referenced by:  ovmpoga  5971  iseqovex  10391  seqvalcd  10394  resqrexlemp1rp  10948  resqrexlemfp1  10951  lcmval  11995  ennnfonelemg  12336  plusfvalg  12594  cnfval  12834  cnpfval  12835  blvalps  13028  blval  13029