Theorem ovmpodx 6147

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 1576	. 2 |- F/ x ph
8		nfv 1576	. 2 |- F/ y ph
9		nfcv 2374	. 2 |- F/_ y A
10		nfcv 2374	. 2 |- F/_ x B
11		nfcv 2374	. 2 |- F/_ x S
12		nfcv 2374	. 2 |- F/_ y S
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	ovmpodxf 6146	1 |- ( ph -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202  (class class class)co 6017   e. cmpo 6019

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022

This theorem is referenced by:  ovmpod  6148  ovmpox  6149  dvfvalap  15404