Theorem ovmpodx 6205

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 1581	. 2 |- F/ x ph
8		nfv 1581	. 2 |- F/ y ph
9		nfcv 2392	. 2 |- F/_ y A
10		nfcv 2392	. 2 |- F/_ x B
11		nfcv 2392	. 2 |- F/_ x S
12		nfcv 2392	. 2 |- F/_ y S
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	ovmpodxf 6204	1 |- ( ph -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209  (class class class)co 6075   e. cmpo 6077

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080

This theorem is referenced by:  ovmpod  6206  ovmpox  6207  dvfvalap  15705