Theorem mpt2fvex 5849

Description: Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)

Hypothesis

Ref	Expression
fmpt2.1	|- F = ( x e. A , y e. B |-> C )

Assertion

Ref	Expression
mpt2fvex	|- ( ( A. x A. y C e. V /\ R e. W /\ S e. X ) -> ( R F S ) e. _V )

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

Allowed substitution hints:   C( x, y)   R( x, y)   S( x, y)   F( x, y)   V( x, y)   W( x, y)   X( x, y)

Proof of Theorem mpt2fvex

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 5535	. 2 |- ( R F S ) = ( F ` <. R , S >. )
2		elex 2610	. . . . . . . . 9 |- ( C e. V -> C e. _V )
3	2	alimi 1384	. . . . . . . 8 |- ( A. y C e. V -> A. y C e. _V )
4		vex 2604	. . . . . . . . 9 |- z e. _V
5		2ndexg 5815	. . . . . . . . 9 |- ( z e. _V -> ( 2nd ` z ) e. _V )
6		nfcv 2219	. . . . . . . . . 10 |- F/_ y ( 2nd ` z )
7		nfcsb1v 2938	. . . . . . . . . . 11 |- F/_ y [_ ( 2nd ` z ) / y ]_ C
8	7	nfel1 2229	. . . . . . . . . 10 |- F/ y [_ ( 2nd ` z ) / y ]_ C e. _V
9		csbeq1a 2916	. . . . . . . . . . 11 |- ( y = ( 2nd ` z ) -> C = [_ ( 2nd ` z ) / y ]_ C )
10	9	eleq1d 2147	. . . . . . . . . 10 |- ( y = ( 2nd ` z ) -> ( C e. _V <-> [_ ( 2nd ` z ) / y ]_ C e. _V ) )
11	6, 8, 10	spcgf 2680	. . . . . . . . 9 |- ( ( 2nd ` z ) e. _V -> ( A. y C e. _V -> [_ ( 2nd ` z ) / y ]_ C e. _V ) )
12	4, 5, 11	mp2b 8	. . . . . . . 8 |- ( A. y C e. _V -> [_ ( 2nd ` z ) / y ]_ C e. _V )
13	3, 12	syl 14	. . . . . . 7 |- ( A. y C e. V -> [_ ( 2nd ` z ) / y ]_ C e. _V )
14	13	alimi 1384	. . . . . 6 |- ( A. x A. y C e. V -> A. x [_ ( 2nd ` z ) / y ]_ C e. _V )
15		1stexg 5814	. . . . . . 7 |- ( z e. _V -> ( 1st ` z ) e. _V )
16		nfcv 2219	. . . . . . . 8 |- F/_ x ( 1st ` z )
17		nfcsb1v 2938	. . . . . . . . 9 |- F/_ x [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C
18	17	nfel1 2229	. . . . . . . 8 |- F/ x [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V
19		csbeq1a 2916	. . . . . . . . 9 |- ( x = ( 1st ` z ) -> [_ ( 2nd ` z ) / y ]_ C = [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C )
20	19	eleq1d 2147	. . . . . . . 8 |- ( x = ( 1st ` z ) -> ( [_ ( 2nd ` z ) / y ]_ C e. _V <-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V ) )
21	16, 18, 20	spcgf 2680	. . . . . . 7 |- ( ( 1st ` z ) e. _V -> ( A. x [_ ( 2nd ` z ) / y ]_ C e. _V -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V ) )
22	4, 15, 21	mp2b 8	. . . . . 6 |- ( A. x [_ ( 2nd ` z ) / y ]_ C e. _V -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V )
23	14, 22	syl 14	. . . . 5 |- ( A. x A. y C e. V -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V )
24	23	alrimiv 1795	. . . 4 |- ( A. x A. y C e. V -> A. z [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V )
25	24	3ad2ant1 959	. . 3 |- ( ( A. x A. y C e. V /\ R e. W /\ S e. X ) -> A. z [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V )
26		opexg 3983	. . . 4 |- ( ( R e. W /\ S e. X ) -> <. R , S >. e. _V )
27	26	3adant1 956	. . 3 |- ( ( A. x A. y C e. V /\ R e. W /\ S e. X ) -> <. R , S >. e. _V )
28		fmpt2.1	. . . . 5 |- F = ( x e. A , y e. B |-> C )
29		mpt2mptsx 5843	. . . . 5 |- ( x e. A , y e. B |-> C ) = ( z e. U_ x e. A ( { x } X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C )
30	28, 29	eqtri 2101	. . . 4 |- F = ( z e. U_ x e. A ( { x } X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C )
31	30	mptfvex 5277	. . 3 |- ( ( A. z [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V /\ <. R , S >. e. _V ) -> ( F ` <. R , S >. ) e. _V )
32	25, 27, 31	syl2anc 403	. 2 |- ( ( A. x A. y C e. V /\ R e. W /\ S e. X ) -> ( F ` <. R , S >. ) e. _V )
33	1, 32	syl5eqel 2165	1 |- ( ( A. x A. y C e. V /\ R e. W /\ S e. X ) -> ( R F S ) e. _V )

Syntax hints:   -> wi 4   /\ w3a 919   A.wal 1282   = wceq 1284   e. wcel 1433   _Vcvv 2601   [_csb 2908   {csn 3398   <.cop 3401   U_ciun 3678   |-> cmpt 3839   X. cxp 4361   ` cfv 4922  (class class class)co 5532   |-> cmpt2 5534   1stc1st 5785   2ndc2nd 5786

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-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fo 4928  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788

This theorem is referenced by:  mpt2fvexi  5852  oaexg  6051  omexg  6054  oeiexg  6056