Theorem ovmpt2s 5649

Description: Value of a function given by the "maps to" notation, expressed using explicit substitution. (Contributed by Mario Carneiro, 30-Apr-2015.)

Hypothesis

Ref	Expression
ovmpt2s.3	|- F = ( x e. C , y e. D |-> R )

Assertion

Ref	Expression
ovmpt2s	|- ( ( A e. C /\ B e. D /\ [_ A / x ]_ [_ B / y ]_ R e. V ) -> ( A F B ) = [_ A / x ]_ [_ B / y ]_ R )

Distinct variable groups:   x, y, A   x, B, y   x, C, y   x, D, y

Allowed substitution hints:   R( x, y)   F( x, y)   V( x, y)

Proof of Theorem ovmpt2s

Step	Hyp	Ref	Expression
1		elex 2611	. . 3 |- ( [_ A / x ]_ [_ B / y ]_ R e. V -> [_ A / x ]_ [_ B / y ]_ R e. _V )
2		nfcv 2220	. . . . 5 |- F/_ x A
3		nfcv 2220	. . . . 5 |- F/_ y A
4		nfcv 2220	. . . . 5 |- F/_ y B
5		nfcsb1v 2939	. . . . . . 7 |- F/_ x [_ A / x ]_ R
6	5	nfel1 2230	. . . . . 6 |- F/ x [_ A / x ]_ R e. _V
7		ovmpt2s.3	. . . . . . . . 9 |- F = ( x e. C , y e. D |-> R )
8		nfmpt21 5596	. . . . . . . . 9 |- F/_ x ( x e. C , y e. D |-> R )
9	7, 8	nfcxfr 2217	. . . . . . . 8 |- F/_ x F
10		nfcv 2220	. . . . . . . 8 |- F/_ x y
11	2, 9, 10	nfov 5560	. . . . . . 7 |- F/_ x ( A F y )
12	11, 5	nfeq 2227	. . . . . 6 |- F/ x ( A F y ) = [_ A / x ]_ R
13	6, 12	nfim 1505	. . . . 5 |- F/ x ( [_ A / x ]_ R e. _V -> ( A F y ) = [_ A / x ]_ R )
14		nfcsb1v 2939	. . . . . . 7 |- F/_ y [_ B / y ]_ [_ A / x ]_ R
15	14	nfel1 2230	. . . . . 6 |- F/ y [_ B / y ]_ [_ A / x ]_ R e. _V
16		nfmpt22 5597	. . . . . . . . 9 |- F/_ y ( x e. C , y e. D |-> R )
17	7, 16	nfcxfr 2217	. . . . . . . 8 |- F/_ y F
18	3, 17, 4	nfov 5560	. . . . . . 7 |- F/_ y ( A F B )
19	18, 14	nfeq 2227	. . . . . 6 |- F/ y ( A F B ) = [_ B / y ]_ [_ A / x ]_ R
20	15, 19	nfim 1505	. . . . 5 |- F/ y ( [_ B / y ]_ [_ A / x ]_ R e. _V -> ( A F B ) = [_ B / y ]_ [_ A / x ]_ R )
21		csbeq1a 2917	. . . . . . 7 |- ( x = A -> R = [_ A / x ]_ R )
22	21	eleq1d 2148	. . . . . 6 |- ( x = A -> ( R e. _V <-> [_ A / x ]_ R e. _V ) )
23		oveq1 5544	. . . . . . 7 |- ( x = A -> ( x F y ) = ( A F y ) )
24	23, 21	eqeq12d 2096	. . . . . 6 |- ( x = A -> ( ( x F y ) = R <-> ( A F y ) = [_ A / x ]_ R ) )
25	22, 24	imbi12d 232	. . . . 5 |- ( x = A -> ( ( R e. _V -> ( x F y ) = R ) <-> ( [_ A / x ]_ R e. _V -> ( A F y ) = [_ A / x ]_ R ) ) )
26		csbeq1a 2917	. . . . . . 7 |- ( y = B -> [_ A / x ]_ R = [_ B / y ]_ [_ A / x ]_ R )
27	26	eleq1d 2148	. . . . . 6 |- ( y = B -> ( [_ A / x ]_ R e. _V <-> [_ B / y ]_ [_ A / x ]_ R e. _V ) )
28		oveq2 5545	. . . . . . 7 |- ( y = B -> ( A F y ) = ( A F B ) )
29	28, 26	eqeq12d 2096	. . . . . 6 |- ( y = B -> ( ( A F y ) = [_ A / x ]_ R <-> ( A F B ) = [_ B / y ]_ [_ A / x ]_ R ) )
30	27, 29	imbi12d 232	. . . . 5 |- ( y = B -> ( ( [_ A / x ]_ R e. _V -> ( A F y ) = [_ A / x ]_ R ) <-> ( [_ B / y ]_ [_ A / x ]_ R e. _V -> ( A F B ) = [_ B / y ]_ [_ A / x ]_ R ) ) )
31	7	ovmpt4g 5648	. . . . . 6 |- ( ( x e. C /\ y e. D /\ R e. _V ) -> ( x F y ) = R )
32	31	3expia 1141	. . . . 5 |- ( ( x e. C /\ y e. D ) -> ( R e. _V -> ( x F y ) = R ) )
33	2, 3, 4, 13, 20, 25, 30, 32	vtocl2gaf 2666	. . . 4 |- ( ( A e. C /\ B e. D ) -> ( [_ B / y ]_ [_ A / x ]_ R e. _V -> ( A F B ) = [_ B / y ]_ [_ A / x ]_ R ) )
34		csbcomg 2930	. . . . 5 |- ( ( A e. C /\ B e. D ) -> [_ A / x ]_ [_ B / y ]_ R = [_ B / y ]_ [_ A / x ]_ R )
35	34	eleq1d 2148	. . . 4 |- ( ( A e. C /\ B e. D ) -> ( [_ A / x ]_ [_ B / y ]_ R e. _V <-> [_ B / y ]_ [_ A / x ]_ R e. _V ) )
36	34	eqeq2d 2093	. . . 4 |- ( ( A e. C /\ B e. D ) -> ( ( A F B ) = [_ A / x ]_ [_ B / y ]_ R <-> ( A F B ) = [_ B / y ]_ [_ A / x ]_ R ) )
37	33, 35, 36	3imtr4d 201	. . 3 |- ( ( A e. C /\ B e. D ) -> ( [_ A / x ]_ [_ B / y ]_ R e. _V -> ( A F B ) = [_ A / x ]_ [_ B / y ]_ R ) )
38	1, 37	syl5 32	. 2 |- ( ( A e. C /\ B e. D ) -> ( [_ A / x ]_ [_ B / y ]_ R e. V -> ( A F B ) = [_ A / x ]_ [_ B / y ]_ R ) )
39	38	3impia 1136	1 |- ( ( A e. C /\ B e. D /\ [_ A / x ]_ [_ B / y ]_ R e. V ) -> ( A F B ) = [_ A / x ]_ [_ B / y ]_ R )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 920   = wceq 1285   e. wcel 1434   _Vcvv 2602   [_csb 2909  (class class class)co 5537   |-> cmpt2 5539

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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-setind 4282

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-iota 4891  df-fun 4928  df-fv 4934  df-ov 5540  df-oprab 5541  df-mpt2 5542

This theorem is referenced by: (None)