Theorem ovmpt2s 6784

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 3212	. . 3 |- ( [_ A / x ]_ [_ B / y ]_ R e. V -> [_ A / x ]_ [_ B / y ]_ R e. _V )
2		nfcv 2764	. . . . 5 |- F/_ x A
3		nfcv 2764	. . . . 5 |- F/_ y A
4		nfcv 2764	. . . . 5 |- F/_ y B
5		nfcsb1v 3549	. . . . . . 7 |- F/_ x [_ A / x ]_ R
6	5	nfel1 2779	. . . . . 6 |- F/ x [_ A / x ]_ R e. _V
7		ovmpt2s.3	. . . . . . . . 9 |- F = ( x e. C , y e. D |-> R )
8		nfmpt21 6722	. . . . . . . . 9 |- F/_ x ( x e. C , y e. D |-> R )
9	7, 8	nfcxfr 2762	. . . . . . . 8 |- F/_ x F
10		nfcv 2764	. . . . . . . 8 |- F/_ x y
11	2, 9, 10	nfov 6676	. . . . . . 7 |- F/_ x ( A F y )
12	11, 5	nfeq 2776	. . . . . 6 |- F/ x ( A F y ) = [_ A / x ]_ R
13	6, 12	nfim 1825	. . . . 5 |- F/ x ( [_ A / x ]_ R e. _V -> ( A F y ) = [_ A / x ]_ R )
14		nfcsb1v 3549	. . . . . . 7 |- F/_ y [_ B / y ]_ [_ A / x ]_ R
15	14	nfel1 2779	. . . . . 6 |- F/ y [_ B / y ]_ [_ A / x ]_ R e. _V
16		nfmpt22 6723	. . . . . . . . 9 |- F/_ y ( x e. C , y e. D |-> R )
17	7, 16	nfcxfr 2762	. . . . . . . 8 |- F/_ y F
18	3, 17, 4	nfov 6676	. . . . . . 7 |- F/_ y ( A F B )
19	18, 14	nfeq 2776	. . . . . 6 |- F/ y ( A F B ) = [_ B / y ]_ [_ A / x ]_ R
20	15, 19	nfim 1825	. . . . 5 |- F/ y ( [_ B / y ]_ [_ A / x ]_ R e. _V -> ( A F B ) = [_ B / y ]_ [_ A / x ]_ R )
21		csbeq1a 3542	. . . . . . 7 |- ( x = A -> R = [_ A / x ]_ R )
22	21	eleq1d 2686	. . . . . 6 |- ( x = A -> ( R e. _V <-> [_ A / x ]_ R e. _V ) )
23		oveq1 6657	. . . . . . 7 |- ( x = A -> ( x F y ) = ( A F y ) )
24	23, 21	eqeq12d 2637	. . . . . 6 |- ( x = A -> ( ( x F y ) = R <-> ( A F y ) = [_ A / x ]_ R ) )
25	22, 24	imbi12d 334	. . . . 5 |- ( x = A -> ( ( R e. _V -> ( x F y ) = R ) <-> ( [_ A / x ]_ R e. _V -> ( A F y ) = [_ A / x ]_ R ) ) )
26		csbeq1a 3542	. . . . . . 7 |- ( y = B -> [_ A / x ]_ R = [_ B / y ]_ [_ A / x ]_ R )
27	26	eleq1d 2686	. . . . . 6 |- ( y = B -> ( [_ A / x ]_ R e. _V <-> [_ B / y ]_ [_ A / x ]_ R e. _V ) )
28		oveq2 6658	. . . . . . 7 |- ( y = B -> ( A F y ) = ( A F B ) )
29	28, 26	eqeq12d 2637	. . . . . 6 |- ( y = B -> ( ( A F y ) = [_ A / x ]_ R <-> ( A F B ) = [_ B / y ]_ [_ A / x ]_ R ) )
30	27, 29	imbi12d 334	. . . . 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 6783	. . . . . 6 |- ( ( x e. C /\ y e. D /\ R e. _V ) -> ( x F y ) = R )
32	31	3expia 1267	. . . . 5 |- ( ( x e. C /\ y e. D ) -> ( R e. _V -> ( x F y ) = R ) )
33	2, 3, 4, 13, 20, 25, 30, 32	vtocl2gaf 3273	. . . 4 |- ( ( A e. C /\ B e. D ) -> ( [_ B / y ]_ [_ A / x ]_ R e. _V -> ( A F B ) = [_ B / y ]_ [_ A / x ]_ R ) )
34		csbcom 3994	. . . . 5 |- [_ A / x ]_ [_ B / y ]_ R = [_ B / y ]_ [_ A / x ]_ R
35	34	eleq1i 2692	. . . 4 |- ( [_ A / x ]_ [_ B / y ]_ R e. _V <-> [_ B / y ]_ [_ A / x ]_ R e. _V )
36	34	eqeq2i 2634	. . . 4 |- ( ( A F B ) = [_ A / x ]_ [_ B / y ]_ R <-> ( A F B ) = [_ B / y ]_ [_ A / x ]_ R )
37	33, 35, 36	3imtr4g 285	. . 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 34	. 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 1261	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 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533  (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-fal 1489  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-csb 3534  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:  finxpreclem2  33227