Theorem ovmpt2s 5624

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 2566	. . 3 |- ( [_ A / x ]_ [_ B / y ]_ R e. V -> [_ A / x ]_ [_ B / y ]_ R e. _V )
2		nfcv 2178	. . . . 5 |- F/_ x A
3		nfcv 2178	. . . . 5 |- F/_ y A
4		nfcv 2178	. . . . 5 |- F/_ y B
5		nfcsb1v 2882	. . . . . . 7 |- F/_ x [_ A / x ]_ R
6	5	nfel1 2188	. . . . . 6 |- F/ x [_ A / x ]_ R e. _V
7		ovmpt2s.3	. . . . . . . . 9 |- F = ( x e. C , y e. D |-> R )
8		nfmpt21 5571	. . . . . . . . 9 |- F/_ x ( x e. C , y e. D |-> R )
9	7, 8	nfcxfr 2175	. . . . . . . 8 |- F/_ x F
10		nfcv 2178	. . . . . . . 8 |- F/_ x y
11	2, 9, 10	nfov 5535	. . . . . . 7 |- F/_ x ( A F y )
12	11, 5	nfeq 2185	. . . . . 6 |- F/ x ( A F y ) = [_ A / x ]_ R
13	6, 12	nfim 1464	. . . . 5 |- F/ x ( [_ A / x ]_ R e. _V -> ( A F y ) = [_ A / x ]_ R )
14		nfcsb1v 2882	. . . . . . 7 |- F/_ y [_ B / y ]_ [_ A / x ]_ R
15	14	nfel1 2188	. . . . . 6 |- F/ y [_ B / y ]_ [_ A / x ]_ R e. _V
16		nfmpt22 5572	. . . . . . . . 9 |- F/_ y ( x e. C , y e. D |-> R )
17	7, 16	nfcxfr 2175	. . . . . . . 8 |- F/_ y F
18	3, 17, 4	nfov 5535	. . . . . . 7 |- F/_ y ( A F B )
19	18, 14	nfeq 2185	. . . . . 6 |- F/ y ( A F B ) = [_ B / y ]_ [_ A / x ]_ R
20	15, 19	nfim 1464	. . . . 5 |- F/ y ( [_ B / y ]_ [_ A / x ]_ R e. _V -> ( A F B ) = [_ B / y ]_ [_ A / x ]_ R )
21		csbeq1a 2860	. . . . . . 7 |- ( x = A -> R = [_ A / x ]_ R )
22	21	eleq1d 2106	. . . . . 6 |- ( x = A -> ( R e. _V <-> [_ A / x ]_ R e. _V ) )
23		oveq1 5519	. . . . . . 7 |- ( x = A -> ( x F y ) = ( A F y ) )
24	23, 21	eqeq12d 2054	. . . . . 6 |- ( x = A -> ( ( x F y ) = R <-> ( A F y ) = [_ A / x ]_ R ) )
25	22, 24	imbi12d 223	. . . . 5 |- ( x = A -> ( ( R e. _V -> ( x F y ) = R ) <-> ( [_ A / x ]_ R e. _V -> ( A F y ) = [_ A / x ]_ R ) ) )
26		csbeq1a 2860	. . . . . . 7 |- ( y = B -> [_ A / x ]_ R = [_ B / y ]_ [_ A / x ]_ R )
27	26	eleq1d 2106	. . . . . 6 |- ( y = B -> ( [_ A / x ]_ R e. _V <-> [_ B / y ]_ [_ A / x ]_ R e. _V ) )
28		oveq2 5520	. . . . . . 7 |- ( y = B -> ( A F y ) = ( A F B ) )
29	28, 26	eqeq12d 2054	. . . . . 6 |- ( y = B -> ( ( A F y ) = [_ A / x ]_ R <-> ( A F B ) = [_ B / y ]_ [_ A / x ]_ R ) )
30	27, 29	imbi12d 223	. . . . 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 5623	. . . . . 6 |- ( ( x e. C /\ y e. D /\ R e. _V ) -> ( x F y ) = R )
32	31	3expia 1106	. . . . 5 |- ( ( x e. C /\ y e. D ) -> ( R e. _V -> ( x F y ) = R ) )
33	2, 3, 4, 13, 20, 25, 30, 32	vtocl2gaf 2620	. . . 4 |- ( ( A e. C /\ B e. D ) -> ( [_ B / y ]_ [_ A / x ]_ R e. _V -> ( A F B ) = [_ B / y ]_ [_ A / x ]_ R ) )
34		csbcomg 2873	. . . . 5 |- ( ( A e. C /\ B e. D ) -> [_ A / x ]_ [_ B / y ]_ R = [_ B / y ]_ [_ A / x ]_ R )
35	34	eleq1d 2106	. . . 4 |- ( ( A e. C /\ B e. D ) -> ( [_ A / x ]_ [_ B / y ]_ R e. _V <-> [_ B / y ]_ [_ A / x ]_ R e. _V ) )
36	34	eqeq2d 2051	. . . 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 192	. . 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 28	. 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 1101	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 97   /\ w3a 885   = wceq 1243   e. wcel 1393   _Vcvv 2557   [_csb 2852  (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-csb 2853  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: (None)