Theorem ovmpt2dxf 6786

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 6667	. 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 2622	. . . . . . . . 9 |- ( x e. C , y e. D |-> R ) = ( x e. C , y e. D |-> R )
8	7	ovmpt4g 6783	. . . . . . . 8 |- ( ( x e. C /\ y e. D /\ R e. _V ) -> ( x ( x e. C , y e. D |-> R ) y ) = R )
9	8	a1i 11	. . . . . . 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 2082	. . . . . 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 3449	. . . . 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 2082	. . . 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 3449	. . 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 481	. . . . 5 |- ( ( ph /\ x = A ) -> B e. L )
15		simplr 792	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> x = A )
16	3	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> A e. C )
17	15, 16	eqeltrd 2701	. . . . . . 7 |- ( ( ( ph /\ x = A ) /\ y = B ) -> x e. C )
18	5	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> B e. L )
19		simpr 477	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> y = B )
20		ovmpt2dx.3	. . . . . . . . 9 |- ( ( ph /\ x = A ) -> D = L )
21	20	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> D = L )
22	18, 19, 21	3eltr4d 2716	. . . . . . 7 |- ( ( ( ph /\ x = A ) /\ y = B ) -> y e. D )
23		ovmpt2dx.2	. . . . . . . . 9 |- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S )
24	23	anassrs 680	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> R = S )
25		ovmpt2dx.6	. . . . . . . . . 10 |- ( ph -> S e. X )
26		elex 3212	. . . . . . . . . 10 |- ( S e. X -> S e. _V )
27	25, 26	syl 17	. . . . . . . . 9 |- ( ph -> S e. _V )
28	27	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> S e. _V )
29	24, 28	eqeltrd 2701	. . . . . . 7 |- ( ( ( ph /\ x = A ) /\ y = B ) -> R e. _V )
30		biimt 350	. . . . . . 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 1326	. . . . . 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 6668	. . . . . . 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 2637	. . . . . 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 270	. . . . 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 2780	. . . . . 6 |- F/ y x = A
37	6, 36	nfan 1828	. . . . 5 |- F/ y ( ph /\ x = A )
38		nfmpt22 6723	. . . . . . . 8 |- F/_ y ( x e. C , y e. D |-> R )
39		nfcv 2764	. . . . . . . 8 |- F/_ y B
40	35, 38, 39	nfov 6676	. . . . . . 7 |- F/_ y ( A ( x e. C , y e. D |-> R ) B )
41		ovmpt2dxf.sy	. . . . . . 7 |- F/_ y S
42	40, 41	nfeq 2776	. . . . . 6 |- F/ y ( A ( x e. C , y e. D |-> R ) B ) = S
43	42	a1i 11	. . . . 5 |- ( ( ph /\ x = A ) -> F/ y ( A ( x e. C , y e. D |-> R ) B ) = S )
44	14, 34, 37, 43	sbciedf 3471	. . . 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 2764	. . . . . . 7 |- F/_ x A
46		nfmpt21 6722	. . . . . . 7 |- F/_ x ( x e. C , y e. D |-> R )
47		ovmpt2dxf.bx	. . . . . . 7 |- F/_ x B
48	45, 46, 47	nfov 6676	. . . . . 6 |- F/_ x ( A ( x e. C , y e. D |-> R ) B )
49		ovmpt2dxf.sx	. . . . . 6 |- F/_ x S
50	48, 49	nfeq 2776	. . . . 5 |- F/ x ( A ( x e. C , y e. D |-> R ) B ) = S
51	50	a1i 11	. . . 4 |- ( ph -> F/ x ( A ( x e. C , y e. D |-> R ) B ) = S )
52	3, 44, 4, 51	sbciedf 3471	. . 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 222	. 2 |- ( ph -> ( A ( x e. C , y e. D |-> R ) B ) = S )
54	2, 53	eqtrd 2656	1 |- ( ph -> ( A F B ) = S )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200   [.wsbc 3435  (class class class)co 6650   |-> cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  ovmpt2dx  6787  elovmpt2rab  6880  elovmpt2rab1  6881  ovmpt3rab1  6891  mpt2xopoveq  7345  fvmpt2curryd  7397  mdetralt2  20415  mdetunilem2  20419  gsummatr01lem4  20464