Theorem ovmpos 6069

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 2783	. . 3 |- ( [_ A / x ]_ [_ B / y ]_ R e. V -> [_ A / x ]_ [_ B / y ]_ R e. _V )
2		nfcv 2348	. . . . 5 |- F/_ x A
3		nfcv 2348	. . . . 5 |- F/_ y A
4		nfcv 2348	. . . . 5 |- F/_ y B
5		nfcsb1v 3126	. . . . . . 7 |- F/_ x [_ A / x ]_ R
6	5	nfel1 2359	. . . . . 6 |- F/ x [_ A / x ]_ R e. _V
7		ovmpos.3	. . . . . . . . 9 |- F = ( x e. C , y e. D |-> R )
8		nfmpo1 6012	. . . . . . . . 9 |- F/_ x ( x e. C , y e. D |-> R )
9	7, 8	nfcxfr 2345	. . . . . . . 8 |- F/_ x F
10		nfcv 2348	. . . . . . . 8 |- F/_ x y
11	2, 9, 10	nfov 5974	. . . . . . 7 |- F/_ x ( A F y )
12	11, 5	nfeq 2356	. . . . . 6 |- F/ x ( A F y ) = [_ A / x ]_ R
13	6, 12	nfim 1595	. . . . 5 |- F/ x ( [_ A / x ]_ R e. _V -> ( A F y ) = [_ A / x ]_ R )
14		nfcsb1v 3126	. . . . . . 7 |- F/_ y [_ B / y ]_ [_ A / x ]_ R
15	14	nfel1 2359	. . . . . 6 |- F/ y [_ B / y ]_ [_ A / x ]_ R e. _V
16		nfmpo2 6013	. . . . . . . . 9 |- F/_ y ( x e. C , y e. D |-> R )
17	7, 16	nfcxfr 2345	. . . . . . . 8 |- F/_ y F
18	3, 17, 4	nfov 5974	. . . . . . 7 |- F/_ y ( A F B )
19	18, 14	nfeq 2356	. . . . . 6 |- F/ y ( A F B ) = [_ B / y ]_ [_ A / x ]_ R
20	15, 19	nfim 1595	. . . . 5 |- F/ y ( [_ B / y ]_ [_ A / x ]_ R e. _V -> ( A F B ) = [_ B / y ]_ [_ A / x ]_ R )
21		csbeq1a 3102	. . . . . . 7 |- ( x = A -> R = [_ A / x ]_ R )
22	21	eleq1d 2274	. . . . . 6 |- ( x = A -> ( R e. _V <-> [_ A / x ]_ R e. _V ) )
23		oveq1 5951	. . . . . . 7 |- ( x = A -> ( x F y ) = ( A F y ) )
24	23, 21	eqeq12d 2220	. . . . . 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 3102	. . . . . . 7 |- ( y = B -> [_ A / x ]_ R = [_ B / y ]_ [_ A / x ]_ R )
27	26	eleq1d 2274	. . . . . 6 |- ( y = B -> ( [_ A / x ]_ R e. _V <-> [_ B / y ]_ [_ A / x ]_ R e. _V ) )
28		oveq2 5952	. . . . . . 7 |- ( y = B -> ( A F y ) = ( A F B ) )
29	28, 26	eqeq12d 2220	. . . . . 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 6068	. . . . . 6 |- ( ( x e. C /\ y e. D /\ R e. _V ) -> ( x F y ) = R )
32	31	3expia 1208	. . . . 5 |- ( ( x e. C /\ y e. D ) -> ( R e. _V -> ( x F y ) = R ) )
33	2, 3, 4, 13, 20, 25, 30, 32	vtocl2gaf 2840	. . . 4 |- ( ( A e. C /\ B e. D ) -> ( [_ B / y ]_ [_ A / x ]_ R e. _V -> ( A F B ) = [_ B / y ]_ [_ A / x ]_ R ) )
34		csbcomg 3116	. . . . 5 |- ( ( A e. C /\ B e. D ) -> [_ A / x ]_ [_ B / y ]_ R = [_ B / y ]_ [_ A / x ]_ R )
35	34	eleq1d 2274	. . . 4 |- ( ( A e. C /\ B e. D ) -> ( [_ A / x ]_ [_ B / y ]_ R e. _V <-> [_ B / y ]_ [_ A / x ]_ R e. _V ) )
36	34	eqeq2d 2217	. . . 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 1203	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 981   = wceq 1373   e. wcel 2176   _Vcvv 2772   [_csb 3093  (class class class)co 5944   e. cmpo 5946

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949

This theorem is referenced by: (None)