Theorem ovmpodxf 5896

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 )
ovmpodxf.px	|- F/ x ph
ovmpodxf.py	|- F/ y ph
ovmpodxf.ay	|- F/_ y A
ovmpodxf.bx	|- F/_ x B
ovmpodxf.sx	|- F/_ x S
ovmpodxf.sy	|- F/_ y S

Assertion

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

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

Allowed substitution hints:   ph( x, y)   A( y)   B( x)   C( x, y)   D( x, y)   R( x, y)   S( x, y)   F( x, y)   L( x, y)   X( x, y)

Proof of Theorem ovmpodxf

Step	Hyp	Ref	Expression
1		ovmpodx.1	. . 3 |- ( ph -> F = ( x e. C , y e. D |-> R ) )
2	1	oveqd 5791	. 2 |- ( ph -> ( A F B ) = ( A ( x e. C , y e. D |-> R ) B ) )
3		ovmpodx.4	. . . 4 |- ( ph -> A e. C )
4		ovmpodxf.px	. . . . 5 |- F/ x ph
5		ovmpodx.5	. . . . . 6 |- ( ph -> B e. L )
6		ovmpodxf.py	. . . . . . 7 |- F/ y ph
7		eqid 2139	. . . . . . . . 9 |- ( x e. C , y e. D |-> R ) = ( x e. C , y e. D |-> R )
8	7	ovmpt4g 5893	. . . . . . . 8 |- ( ( x e. C /\ y e. D /\ R e. _V ) -> ( x ( x e. C , y e. D |-> R ) y ) = R )
9	8	a1i 9	. . . . . . 7 |- ( ph -> ( ( x e. C /\ y e. D /\ R e. _V ) -> ( x ( x e. C , y e. D |-> R ) y ) = R ) )
10	6, 9	alrimi 1502	. . . . . 6 |- ( ph -> A. y ( ( x e. C /\ y e. D /\ R e. _V ) -> ( x ( x e. C , y e. D |-> R ) y ) = R ) )
11	5, 10	spsbcd 2921	. . . . 5 |- ( ph -> [. B / y ]. ( ( x e. C /\ y e. D /\ R e. _V ) -> ( x ( x e. C , y e. D |-> R ) y ) = R ) )
12	4, 11	alrimi 1502	. . . 4 |- ( ph -> A. x [. B / y ]. ( ( x e. C /\ y e. D /\ R e. _V ) -> ( x ( x e. C , y e. D |-> R ) y ) = R ) )
13	3, 12	spsbcd 2921	. . 3 |- ( ph -> [. A / x ]. [. B / y ]. ( ( x e. C /\ y e. D /\ R e. _V ) -> ( x ( x e. C , y e. D |-> R ) y ) = R ) )
14	5	adantr 274	. . . . 5 |- ( ( ph /\ x = A ) -> B e. L )
15		simplr 519	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> x = A )
16	3	ad2antrr 479	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> A e. C )
17	15, 16	eqeltrd 2216	. . . . . . 7 |- ( ( ( ph /\ x = A ) /\ y = B ) -> x e. C )
18	5	ad2antrr 479	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> B e. L )
19		simpr 109	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> y = B )
20		ovmpodx.3	. . . . . . . . 9 |- ( ( ph /\ x = A ) -> D = L )
21	20	adantr 274	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> D = L )
22	18, 19, 21	3eltr4d 2223	. . . . . . 7 |- ( ( ( ph /\ x = A ) /\ y = B ) -> y e. D )
23		ovmpodx.2	. . . . . . . . 9 |- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S )
24	23	anassrs 397	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> R = S )
25		ovmpodx.6	. . . . . . . . . 10 |- ( ph -> S e. X )
26		elex 2697	. . . . . . . . . 10 |- ( S e. X -> S e. _V )
27	25, 26	syl 14	. . . . . . . . 9 |- ( ph -> S e. _V )
28	27	ad2antrr 479	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> S e. _V )
29	24, 28	eqeltrd 2216	. . . . . . 7 |- ( ( ( ph /\ x = A ) /\ y = B ) -> R e. _V )
30		biimt 240	. . . . . . 7 |- ( ( x e. C /\ y e. D /\ R e. _V ) -> ( ( x ( x e. C , y e. D |-> R ) y ) = R <-> ( ( x e. C /\ y e. D /\ R e. _V ) -> ( x ( x e. C , y e. D |-> R ) y ) = R ) ) )
31	17, 22, 29, 30	syl3anc 1216	. . . . . 6 |- ( ( ( ph /\ x = A ) /\ y = B ) -> ( ( x ( x e. C , y e. D |-> R ) y ) = R <-> ( ( x e. C /\ y e. D /\ R e. _V ) -> ( x ( x e. C , y e. D |-> R ) y ) = R ) ) )
32	15, 19	oveq12d 5792	. . . . . . 7 |- ( ( ( ph /\ x = A ) /\ y = B ) -> ( x ( x e. C , y e. D |-> R ) y ) = ( A ( x e. C , y e. D |-> R ) B ) )
33	32, 24	eqeq12d 2154	. . . . . 6 |- ( ( ( ph /\ x = A ) /\ y = B ) -> ( ( x ( x e. C , y e. D |-> R ) y ) = R <-> ( A ( x e. C , y e. D |-> R ) B ) = S ) )
34	31, 33	bitr3d 189	. . . . 5 |- ( ( ( ph /\ x = A ) /\ y = B ) -> ( ( ( x e. C /\ y e. D /\ R e. _V ) -> ( x ( x e. C , y e. D |-> R ) y ) = R ) <-> ( A ( x e. C , y e. D |-> R ) B ) = S ) )
35		ovmpodxf.ay	. . . . . . 7 |- F/_ y A
36	35	nfeq2 2293	. . . . . 6 |- F/ y x = A
37	6, 36	nfan 1544	. . . . 5 |- F/ y ( ph /\ x = A )
38		nfmpo2 5839	. . . . . . . 8 |- F/_ y ( x e. C , y e. D |-> R )
39		nfcv 2281	. . . . . . . 8 |- F/_ y B
40	35, 38, 39	nfov 5801	. . . . . . 7 |- F/_ y ( A ( x e. C , y e. D |-> R ) B )
41		ovmpodxf.sy	. . . . . . 7 |- F/_ y S
42	40, 41	nfeq 2289	. . . . . 6 |- F/ y ( A ( x e. C , y e. D |-> R ) B ) = S
43	42	a1i 9	. . . . 5 |- ( ( ph /\ x = A ) -> F/ y ( A ( x e. C , y e. D |-> R ) B ) = S )
44	14, 34, 37, 43	sbciedf 2944	. . . 4 |- ( ( ph /\ x = A ) -> ( [. B / y ]. ( ( x e. C /\ y e. D /\ R e. _V ) -> ( x ( x e. C , y e. D |-> R ) y ) = R ) <-> ( A ( x e. C , y e. D |-> R ) B ) = S ) )
45		nfcv 2281	. . . . . . 7 |- F/_ x A
46		nfmpo1 5838	. . . . . . 7 |- F/_ x ( x e. C , y e. D |-> R )
47		ovmpodxf.bx	. . . . . . 7 |- F/_ x B
48	45, 46, 47	nfov 5801	. . . . . 6 |- F/_ x ( A ( x e. C , y e. D |-> R ) B )
49		ovmpodxf.sx	. . . . . 6 |- F/_ x S
50	48, 49	nfeq 2289	. . . . 5 |- F/ x ( A ( x e. C , y e. D |-> R ) B ) = S
51	50	a1i 9	. . . 4 |- ( ph -> F/ x ( A ( x e. C , y e. D |-> R ) B ) = S )
52	3, 44, 4, 51	sbciedf 2944	. . 3 |- ( ph -> ( [. A / x ]. [. B / y ]. ( ( x e. C /\ y e. D /\ R e. _V ) -> ( x ( x e. C , y e. D |-> R ) y ) = R ) <-> ( A ( x e. C , y e. D |-> R ) B ) = S ) )
53	13, 52	mpbid 146	. 2 |- ( ph -> ( A ( x e. C , y e. D |-> R ) B ) = S )
54	2, 53	eqtrd 2172	1 |- ( ph -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   F/wnf 1436   e. wcel 1480   F/_wnfc 2268   _Vcvv 2686   [.wsbc 2909  (class class class)co 5774   e. cmpo 5776

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779

This theorem is referenced by:  ovmpodx  5897  mpoxopoveq  6137