Theorem ovmpt2dxf 5626

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 5529	. 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 2040	. . . . . . . . 9 |- ( x e. C , y e. D |-> R ) = ( x e. C , y e. D |-> R )
8	7	ovmpt4g 5623	. . . . . . . 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 1415	. . . . . 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 2776	. . . . 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 1415	. . . 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 2776	. . 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 261	. . . . 5 |- ( ( ph /\ x = A ) -> B e. L )
15		simplr 482	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> x = A )
16	3	ad2antrr 457	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> A e. C )
17	15, 16	eqeltrd 2114	. . . . . . 7 |- ( ( ( ph /\ x = A ) /\ y = B ) -> x e. C )
18	5	ad2antrr 457	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> B e. L )
19		simpr 103	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> y = B )
20		ovmpt2dx.3	. . . . . . . . 9 |- ( ( ph /\ x = A ) -> D = L )
21	20	adantr 261	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> D = L )
22	18, 19, 21	3eltr4d 2121	. . . . . . 7 |- ( ( ( ph /\ x = A ) /\ y = B ) -> y e. D )
23		ovmpt2dx.2	. . . . . . . . 9 |- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S )
24	23	anassrs 380	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> R = S )
25		ovmpt2dx.6	. . . . . . . . . 10 |- ( ph -> S e. X )
26		elex 2566	. . . . . . . . . 10 |- ( S e. X -> S e. _V )
27	25, 26	syl 14	. . . . . . . . 9 |- ( ph -> S e. _V )
28	27	ad2antrr 457	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> S e. _V )
29	24, 28	eqeltrd 2114	. . . . . . 7 |- ( ( ( ph /\ x = A ) /\ y = B ) -> R e. _V )
30		biimt 230	. . . . . . 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 1135	. . . . . 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 5530	. . . . . . 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 2054	. . . . . 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 179	. . . . 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 2189	. . . . . 6 |- F/ y x = A
37	6, 36	nfan 1457	. . . . 5 |- F/ y ( ph /\ x = A )
38		nfmpt22 5572	. . . . . . . 8 |- F/_ y ( x e. C , y e. D |-> R )
39		nfcv 2178	. . . . . . . 8 |- F/_ y B
40	35, 38, 39	nfov 5535	. . . . . . 7 |- F/_ y ( A ( x e. C , y e. D |-> R ) B )
41		ovmpt2dxf.sy	. . . . . . 7 |- F/_ y S
42	40, 41	nfeq 2185	. . . . . 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 2798	. . . 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 2178	. . . . . . 7 |- F/_ x A
46		nfmpt21 5571	. . . . . . 7 |- F/_ x ( x e. C , y e. D |-> R )
47		ovmpt2dxf.bx	. . . . . . 7 |- F/_ x B
48	45, 46, 47	nfov 5535	. . . . . 6 |- F/_ x ( A ( x e. C , y e. D |-> R ) B )
49		ovmpt2dxf.sx	. . . . . 6 |- F/_ x S
50	48, 49	nfeq 2185	. . . . 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 2798	. . 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 135	. 2 |- ( ph -> ( A ( x e. C , y e. D |-> R ) B ) = S )
54	2, 53	eqtrd 2072	1 |- ( ph -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 885   = wceq 1243   F/wnf 1349   e. wcel 1393   F/_wnfc 2165   _Vcvv 2557   [.wsbc 2764  (class class class)co 5512   |-> cmpt2 5514

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517

This theorem is referenced by:  ovmpt2dx  5627  mpt2xopoveq  5855