Theorem ovmpos 6202

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

Assertion

Ref	Expression
ovmpos	|- ( ( 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 ovmpos

Step	Hyp	Ref	Expression
1		elex 2833	. . 3 |- ( [_ A / x ]_ [_ B / y ]_ R e. V -> [_ A / x ]_ [_ B / y ]_ R e. _V )
2		nfcv 2392	. . . . 5 |- F/_ x A
3		nfcv 2392	. . . . 5 |- F/_ y A
4		nfcv 2392	. . . . 5 |- F/_ y B
5		nfcsb1v 3180	. . . . . . 7 |- F/_ x [_ A / x ]_ R
6	5	nfel1 2403	. . . . . 6 |- F/ x [_ A / x ]_ R e. _V
7		ovmpos.3	. . . . . . . . 9 |- F = ( x e. C , y e. D |-> R )
8		nfmpo1 6145	. . . . . . . . 9 |- F/_ x ( x e. C , y e. D |-> R )
9	7, 8	nfcxfr 2389	. . . . . . . 8 |- F/_ x F
10		nfcv 2392	. . . . . . . 8 |- F/_ x y
11	2, 9, 10	nfov 6105	. . . . . . 7 |- F/_ x ( A F y )
12	11, 5	nfeq 2400	. . . . . 6 |- F/ x ( A F y ) = [_ A / x ]_ R
13	6, 12	nfim 1625	. . . . 5 |- F/ x ( [_ A / x ]_ R e. _V -> ( A F y ) = [_ A / x ]_ R )
14		nfcsb1v 3180	. . . . . . 7 |- F/_ y [_ B / y ]_ [_ A / x ]_ R
15	14	nfel1 2403	. . . . . 6 |- F/ y [_ B / y ]_ [_ A / x ]_ R e. _V
16		nfmpo2 6146	. . . . . . . . 9 |- F/_ y ( x e. C , y e. D |-> R )
17	7, 16	nfcxfr 2389	. . . . . . . 8 |- F/_ y F
18	3, 17, 4	nfov 6105	. . . . . . 7 |- F/_ y ( A F B )
19	18, 14	nfeq 2400	. . . . . 6 |- F/ y ( A F B ) = [_ B / y ]_ [_ A / x ]_ R
20	15, 19	nfim 1625	. . . . 5 |- F/ y ( [_ B / y ]_ [_ A / x ]_ R e. _V -> ( A F B ) = [_ B / y ]_ [_ A / x ]_ R )
21		csbeq1a 3156	. . . . . . 7 |- ( x = A -> R = [_ A / x ]_ R )
22	21	eleq1d 2307	. . . . . 6 |- ( x = A -> ( R e. _V <-> [_ A / x ]_ R e. _V ) )
23		oveq1 6082	. . . . . . 7 |- ( x = A -> ( x F y ) = ( A F y ) )
24	23, 21	eqeq12d 2253	. . . . . 6 |- ( x = A -> ( ( x F y ) = R <-> ( A F y ) = [_ A / x ]_ R ) )
25	22, 24	imbi12d 234	. . . . 5 |- ( x = A -> ( ( R e. _V -> ( x F y ) = R ) <-> ( [_ A / x ]_ R e. _V -> ( A F y ) = [_ A / x ]_ R ) ) )
26		csbeq1a 3156	. . . . . . 7 |- ( y = B -> [_ A / x ]_ R = [_ B / y ]_ [_ A / x ]_ R )
27	26	eleq1d 2307	. . . . . 6 |- ( y = B -> ( [_ A / x ]_ R e. _V <-> [_ B / y ]_ [_ A / x ]_ R e. _V ) )
28		oveq2 6083	. . . . . . 7 |- ( y = B -> ( A F y ) = ( A F B ) )
29	28, 26	eqeq12d 2253	. . . . . 6 |- ( y = B -> ( ( A F y ) = [_ A / x ]_ R <-> ( A F B ) = [_ B / y ]_ [_ A / x ]_ R ) )
30	27, 29	imbi12d 234	. . . . 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 6201	. . . . . 6 |- ( ( x e. C /\ y e. D /\ R e. _V ) -> ( x F y ) = R )
32	31	3expia 1236	. . . . 5 |- ( ( x e. C /\ y e. D ) -> ( R e. _V -> ( x F y ) = R ) )
33	2, 3, 4, 13, 20, 25, 30, 32	vtocl2gaf 2890	. . . 4 |- ( ( A e. C /\ B e. D ) -> ( [_ B / y ]_ [_ A / x ]_ R e. _V -> ( A F B ) = [_ B / y ]_ [_ A / x ]_ R ) )
34		csbcomg 3170	. . . . 5 |- ( ( A e. C /\ B e. D ) -> [_ A / x ]_ [_ B / y ]_ R = [_ B / y ]_ [_ A / x ]_ R )
35	34	eleq1d 2307	. . . 4 |- ( ( A e. C /\ B e. D ) -> ( [_ A / x ]_ [_ B / y ]_ R e. _V <-> [_ B / y ]_ [_ A / x ]_ R e. _V ) )
36	34	eqeq2d 2250	. . . 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 203	. . 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 1231	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 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   _Vcvv 2821   [_csb 3147  (class class class)co 6075   e. cmpo 6077

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080

This theorem is referenced by: (None)