Theorem ovmpt2rdxf 42117

Description: Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpt2dxf 6786. (Contributed by AV, 30-Mar-2019.)

Hypotheses

Ref	Expression
ovmpt2rdx.1	|- ( ph -> F = ( x e. C , y e. D |-> R ) )
ovmpt2rdx.2	|- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S )
ovmpt2rdx.3	|- ( ( ph /\ y = B ) -> C = L )
ovmpt2rdx.4	|- ( ph -> A e. L )
ovmpt2rdx.5	|- ( ph -> B e. D )
ovmpt2rdx.6	|- ( ph -> S e. X )
ovmpt2rdxf.px	|- F/ x ph
ovmpt2rdxf.py	|- F/ y ph
ovmpt2rdxf.ay	|- F/_ y A
ovmpt2rdxf.bx	|- F/_ x B
ovmpt2rdxf.sx	|- F/_ x S
ovmpt2rdxf.sy	|- F/_ y S

Assertion

Ref	Expression
ovmpt2rdxf	|- ( 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 ovmpt2rdxf

Step	Hyp	Ref	Expression
1		ovmpt2rdx.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		ovmpt2rdx.4	. . . 4 |- ( ph -> A e. L )
4		ovmpt2rdxf.px	. . . . 5 |- F/ x ph
5		ovmpt2rdx.5	. . . . . 6 |- ( ph -> B e. D )
6		ovmpt2rdxf.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. X ) -> ( x ( x e. C , y e. D |-> R ) y ) = R )
9	8	a1i 11	. . . . . . 7 |- ( ph -> ( ( x e. C /\ y e. D /\ R e. X ) -> ( 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. X ) -> ( 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. X ) -> ( 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. X ) -> ( 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. X ) -> ( x ( x e. C , y e. D |-> R ) y ) = R ) )
14	5	adantr 481	. . . . 5 |- ( ( ph /\ x = A ) -> B e. D )
15	3	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> A e. L )
16		simpr 477	. . . . . . . . 9 |- ( ( ph /\ x = A ) -> x = A )
17	16	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> x = A )
18		ovmpt2rdx.3	. . . . . . . . 9 |- ( ( ph /\ y = B ) -> C = L )
19	18	adantlr 751	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> C = L )
20	15, 17, 19	3eltr4d 2716	. . . . . . 7 |- ( ( ( ph /\ x = A ) /\ y = B ) -> x e. C )
21	5	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> B e. D )
22		eleq1 2689	. . . . . . . . 9 |- ( y = B -> ( y e. D <-> B e. D ) )
23	22	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> ( y e. D <-> B e. D ) )
24	21, 23	mpbird 247	. . . . . . 7 |- ( ( ( ph /\ x = A ) /\ y = B ) -> y e. D )
25		ovmpt2rdx.2	. . . . . . . . 9 |- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S )
26	25	anassrs 680	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> R = S )
27		ovmpt2rdx.6	. . . . . . . . 9 |- ( ph -> S e. X )
28	27	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> S e. X )
29	26, 28	eqeltrd 2701	. . . . . . 7 |- ( ( ( ph /\ x = A ) /\ y = B ) -> R e. X )
30		biimt 350	. . . . . . 7 |- ( ( x e. C /\ y e. D /\ R e. X ) -> ( ( x ( x e. C , y e. D |-> R ) y ) = R <-> ( ( x e. C /\ y e. D /\ R e. X ) -> ( x ( x e. C , y e. D |-> R ) y ) = R ) ) )
31	20, 24, 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. X ) -> ( x ( x e. C , y e. D |-> R ) y ) = R ) ) )
32		simpr 477	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> y = B )
33	17, 32	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 ) )
34	33, 26	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 ) )
35	31, 34	bitr3d 270	. . . . 5 |- ( ( ( ph /\ x = A ) /\ y = B ) -> ( ( ( x e. C /\ y e. D /\ R e. X ) -> ( x ( x e. C , y e. D |-> R ) y ) = R ) <-> ( A ( x e. C , y e. D |-> R ) B ) = S ) )
36		ovmpt2rdxf.ay	. . . . . . 7 |- F/_ y A
37	36	nfeq2 2780	. . . . . 6 |- F/ y x = A
38	6, 37	nfan 1828	. . . . 5 |- F/ y ( ph /\ x = A )
39		nfmpt22 6723	. . . . . . . 8 |- F/_ y ( x e. C , y e. D |-> R )
40		nfcv 2764	. . . . . . . 8 |- F/_ y B
41	36, 39, 40	nfov 6676	. . . . . . 7 |- F/_ y ( A ( x e. C , y e. D |-> R ) B )
42		ovmpt2rdxf.sy	. . . . . . 7 |- F/_ y S
43	41, 42	nfeq 2776	. . . . . 6 |- F/ y ( A ( x e. C , y e. D |-> R ) B ) = S
44	43	a1i 11	. . . . 5 |- ( ( ph /\ x = A ) -> F/ y ( A ( x e. C , y e. D |-> R ) B ) = S )
45	14, 35, 38, 44	sbciedf 3471	. . . 4 |- ( ( ph /\ x = A ) -> ( [. B / y ]. ( ( x e. C /\ y e. D /\ R e. X ) -> ( x ( x e. C , y e. D |-> R ) y ) = R ) <-> ( A ( x e. C , y e. D |-> R ) B ) = S ) )
46		nfcv 2764	. . . . . . 7 |- F/_ x A
47		nfmpt21 6722	. . . . . . 7 |- F/_ x ( x e. C , y e. D |-> R )
48		ovmpt2rdxf.bx	. . . . . . 7 |- F/_ x B
49	46, 47, 48	nfov 6676	. . . . . 6 |- F/_ x ( A ( x e. C , y e. D |-> R ) B )
50		ovmpt2rdxf.sx	. . . . . 6 |- F/_ x S
51	49, 50	nfeq 2776	. . . . 5 |- F/ x ( A ( x e. C , y e. D |-> R ) B ) = S
52	51	a1i 11	. . . 4 |- ( ph -> F/ x ( A ( x e. C , y e. D |-> R ) B ) = S )
53	3, 45, 4, 52	sbciedf 3471	. . 3 |- ( ph -> ( [. A / x ]. [. B / y ]. ( ( x e. C /\ y e. D /\ R e. X ) -> ( x ( x e. C , y e. D |-> R ) y ) = R ) <-> ( A ( x e. C , y e. D |-> R ) B ) = S ) )
54	13, 53	mpbid 222	. 2 |- ( ph -> ( A ( x e. C , y e. D |-> R ) B ) = S )
55	2, 54	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   [.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:  ovmpt2rdx  42118