Theorem ovmpodx 6049

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 1542	. 2 |- F/ x ph
8		nfv 1542	. 2 |- F/ y ph
9		nfcv 2339	. 2 |- F/_ y A
10		nfcv 2339	. 2 |- F/_ x B
11		nfcv 2339	. 2 |- F/_ x S
12		nfcv 2339	. 2 |- F/_ y S
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	ovmpodxf 6048	1 |- ( ph -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167  (class class class)co 5922   e. cmpo 5924

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927

This theorem is referenced by:  ovmpod  6050  ovmpox  6051  dvfvalap  14917