Theorem ovmpodx 6188

Description: Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)

Hypotheses

Ref	Expression
ovmpodx.1	|- ( ph -> F = ( x e. C , y e. D |-> R ) )
ovmpodx.2	|- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S )
ovmpodx.3	|- ( ( ph /\ x = A ) -> D = L )
ovmpodx.4	|- ( ph -> A e. C )
ovmpodx.5	|- ( ph -> B e. L )
ovmpodx.6	|- ( ph -> S e. X )

Assertion

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

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

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

Proof of Theorem ovmpodx

Step	Hyp	Ref	Expression
1		ovmpodx.1	. 2 |- ( ph -> F = ( x e. C , y e. D |-> R ) )
2		ovmpodx.2	. 2 |- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S )
3		ovmpodx.3	. 2 |- ( ( ph /\ x = A ) -> D = L )
4		ovmpodx.4	. 2 |- ( ph -> A e. C )
5		ovmpodx.5	. 2 |- ( ph -> B e. L )
6		ovmpodx.6	. 2 |- ( ph -> S e. X )
7		nfv 1577	. 2 |- F/ x ph
8		nfv 1577	. 2 |- F/ y ph
9		nfcv 2386	. 2 |- F/_ y A
10		nfcv 2386	. 2 |- F/_ x B
11		nfcv 2386	. 2 |- F/_ x S
12		nfcv 2386	. 2 |- F/_ y S
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	ovmpodxf 6187	1 |- ( ph -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205  (class class class)co 6058   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-in1 619  ax-in2 620  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-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  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-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-dif 3216  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-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063

This theorem is referenced by:  ovmpod  6189  ovmpox  6190  dvfvalap  15672