Theorem ovmpt2dxf 5646

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

Hypotheses

Ref	Expression
ovmpt2dx.1	|- ( ph -> F = ( x e. C , y e. D |-> R ) )
ovmpt2dx.2	|- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S )
ovmpt2dx.3	|- ( ( ph /\ x = A ) -> D = L )
ovmpt2dx.4	|- ( ph -> A e. C )
ovmpt2dx.5	|- ( ph -> B e. L )
ovmpt2dx.6	|- ( ph -> S e. X )
ovmpt2dxf.px	|- F/ x ph
ovmpt2dxf.py	|- F/ y ph
ovmpt2dxf.ay	|- F/_ y A
ovmpt2dxf.bx	|- F/_ x B
ovmpt2dxf.sx	|- F/_ x S
ovmpt2dxf.sy	|- F/_ y S

Assertion

Ref	Expression
ovmpt2dxf	|- ( 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 ovmpt2dxf

Step	Hyp	Ref	Expression
1		ovmpt2dx.1	. . 3 |- ( ph -> F = ( x e. C , y e. D |-> R ) )
2	1	oveqd 5549	. 2 |- ( ph -> ( A F B ) = ( A ( x e. C , y e. D |-> R ) B ) )
3		ovmpt2dx.4	. . . 4 |- ( ph -> A e. C )
4		ovmpt2dxf.px	. . . . 5 |- F/ x ph
5		ovmpt2dx.5	. . . . . 6 |- ( ph -> B e. L )
6		ovmpt2dxf.py	. . . . . . 7 |- F/ y ph
7		eqid 2081	. . . . . . . . 9 |- ( x e. C , y e. D |-> R ) = ( x e. C , y e. D |-> R )
8	7	ovmpt4g 5643	. . . . . . . 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 1455	. . . . . 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 2827	. . . . 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 1455	. . . 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 2827	. . 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 270	. . . . 5 |- ( ( ph /\ x = A ) -> B e. L )
15		simplr 496	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> x = A )
16	3	ad2antrr 471	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> A e. C )
17	15, 16	eqeltrd 2155	. . . . . . 7 |- ( ( ( ph /\ x = A ) /\ y = B ) -> x e. C )
18	5	ad2antrr 471	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> B e. L )
19		simpr 108	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> y = B )
20		ovmpt2dx.3	. . . . . . . . 9 |- ( ( ph /\ x = A ) -> D = L )
21	20	adantr 270	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> D = L )
22	18, 19, 21	3eltr4d 2162	. . . . . . 7 |- ( ( ( ph /\ x = A ) /\ y = B ) -> y e. D )
23		ovmpt2dx.2	. . . . . . . . 9 |- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S )
24	23	anassrs 392	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> R = S )
25		ovmpt2dx.6	. . . . . . . . . 10 |- ( ph -> S e. X )
26		elex 2610	. . . . . . . . . 10 |- ( S e. X -> S e. _V )
27	25, 26	syl 14	. . . . . . . . 9 |- ( ph -> S e. _V )
28	27	ad2antrr 471	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> S e. _V )
29	24, 28	eqeltrd 2155	. . . . . . 7 |- ( ( ( ph /\ x = A ) /\ y = B ) -> R e. _V )
30		biimt 239	. . . . . . 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 1169	. . . . . 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 5550	. . . . . . 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 2095	. . . . . 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 188	. . . . 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		ovmpt2dxf.ay	. . . . . . 7 |- F/_ y A
36	35	nfeq2 2230	. . . . . 6 |- F/ y x = A
37	6, 36	nfan 1497	. . . . 5 |- F/ y ( ph /\ x = A )
38		nfmpt22 5592	. . . . . . . 8 |- F/_ y ( x e. C , y e. D |-> R )
39		nfcv 2219	. . . . . . . 8 |- F/_ y B
40	35, 38, 39	nfov 5555	. . . . . . 7 |- F/_ y ( A ( x e. C , y e. D |-> R ) B )
41		ovmpt2dxf.sy	. . . . . . 7 |- F/_ y S
42	40, 41	nfeq 2226	. . . . . 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 2849	. . . 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 2219	. . . . . . 7 |- F/_ x A
46		nfmpt21 5591	. . . . . . 7 |- F/_ x ( x e. C , y e. D |-> R )
47		ovmpt2dxf.bx	. . . . . . 7 |- F/_ x B
48	45, 46, 47	nfov 5555	. . . . . 6 |- F/_ x ( A ( x e. C , y e. D |-> R ) B )
49		ovmpt2dxf.sx	. . . . . 6 |- F/_ x S
50	48, 49	nfeq 2226	. . . . 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 2849	. . 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 145	. 2 |- ( ph -> ( A ( x e. C , y e. D |-> R ) B ) = S )
54	2, 53	eqtrd 2113	1 |- ( ph -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   F/wnf 1389   e. wcel 1433   F/_wnfc 2206   _Vcvv 2601   [.wsbc 2815  (class class class)co 5532   |-> cmpt2 5534

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537

This theorem is referenced by:  ovmpt2dx  5647  mpt2xopoveq  5878