Theorem ovmpodxf 6044

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 5935	. 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 2193	. . . . . . . . 9 |- ( x e. C , y e. D |-> R ) = ( x e. C , y e. D |-> R )
8	7	ovmpt4g 6041	. . . . . . . 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 1533	. . . . . 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 2998	. . . . 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 1533	. . . 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 2998	. . 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 276	. . . . 5 |- ( ( ph /\ x = A ) -> B e. L )
15		simplr 528	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> x = A )
16	3	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> A e. C )
17	15, 16	eqeltrd 2270	. . . . . . 7 |- ( ( ( ph /\ x = A ) /\ y = B ) -> x e. C )
18	5	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> B e. L )
19		simpr 110	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> y = B )
20		ovmpodx.3	. . . . . . . . 9 |- ( ( ph /\ x = A ) -> D = L )
21	20	adantr 276	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> D = L )
22	18, 19, 21	3eltr4d 2277	. . . . . . 7 |- ( ( ( ph /\ x = A ) /\ y = B ) -> y e. D )
23		ovmpodx.2	. . . . . . . . 9 |- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S )
24	23	anassrs 400	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> R = S )
25		ovmpodx.6	. . . . . . . . . 10 |- ( ph -> S e. X )
26		elex 2771	. . . . . . . . . 10 |- ( S e. X -> S e. _V )
27	25, 26	syl 14	. . . . . . . . 9 |- ( ph -> S e. _V )
28	27	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ x = A ) /\ y = B ) -> S e. _V )
29	24, 28	eqeltrd 2270	. . . . . . 7 |- ( ( ( ph /\ x = A ) /\ y = B ) -> R e. _V )
30		biimt 241	. . . . . . 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 1249	. . . . . 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 5936	. . . . . . 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 2208	. . . . . 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 190	. . . . 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 2348	. . . . . 6 |- F/ y x = A
37	6, 36	nfan 1576	. . . . 5 |- F/ y ( ph /\ x = A )
38		nfmpo2 5986	. . . . . . . 8 |- F/_ y ( x e. C , y e. D |-> R )
39		nfcv 2336	. . . . . . . 8 |- F/_ y B
40	35, 38, 39	nfov 5948	. . . . . . 7 |- F/_ y ( A ( x e. C , y e. D |-> R ) B )
41		ovmpodxf.sy	. . . . . . 7 |- F/_ y S
42	40, 41	nfeq 2344	. . . . . 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 3021	. . . 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 2336	. . . . . . 7 |- F/_ x A
46		nfmpo1 5985	. . . . . . 7 |- F/_ x ( x e. C , y e. D |-> R )
47		ovmpodxf.bx	. . . . . . 7 |- F/_ x B
48	45, 46, 47	nfov 5948	. . . . . 6 |- F/_ x ( A ( x e. C , y e. D |-> R ) B )
49		ovmpodxf.sx	. . . . . 6 |- F/_ x S
50	48, 49	nfeq 2344	. . . . 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 3021	. . 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 147	. 2 |- ( ph -> ( A ( x e. C , y e. D |-> R ) B ) = S )
54	2, 53	eqtrd 2226	1 |- ( ph -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   F/wnf 1471   e. wcel 2164   F/_wnfc 2323   _Vcvv 2760   [.wsbc 2985  (class class class)co 5918   e. cmpo 5920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923

This theorem is referenced by:  ovmpodx  6045  elovmporab  6118  elovmporab1w  6119  mpoxopoveq  6293