Theorem mpofvex 6232

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

Hypothesis

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

Assertion

Ref	Expression
mpofvex	|- ( ( 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 mpofvex

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 5903	. 2 |- ( R F S ) = ( F ` <. R , S >. )
2		elex 2763	. . . . . . . . 9 |- ( C e. V -> C e. _V )
3	2	alimi 1466	. . . . . . . 8 |- ( A. y C e. V -> A. y C e. _V )
4		vex 2755	. . . . . . . . 9 |- z e. _V
5		2ndexg 6197	. . . . . . . . 9 |- ( z e. _V -> ( 2nd ` z ) e. _V )
6		nfcv 2332	. . . . . . . . . 10 |- F/_ y ( 2nd ` z )
7		nfcsb1v 3105	. . . . . . . . . . 11 |- F/_ y [_ ( 2nd ` z ) / y ]_ C
8	7	nfel1 2343	. . . . . . . . . 10 |- F/ y [_ ( 2nd ` z ) / y ]_ C e. _V
9		csbeq1a 3081	. . . . . . . . . . 11 |- ( y = ( 2nd ` z ) -> C = [_ ( 2nd ` z ) / y ]_ C )
10	9	eleq1d 2258	. . . . . . . . . 10 |- ( y = ( 2nd ` z ) -> ( C e. _V <-> [_ ( 2nd ` z ) / y ]_ C e. _V ) )
11	6, 8, 10	spcgf 2834	. . . . . . . . 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 1466	. . . . . 6 |- ( A. x A. y C e. V -> A. x [_ ( 2nd ` z ) / y ]_ C e. _V )
15		1stexg 6196	. . . . . . 7 |- ( z e. _V -> ( 1st ` z ) e. _V )
16		nfcv 2332	. . . . . . . 8 |- F/_ x ( 1st ` z )
17		nfcsb1v 3105	. . . . . . . . 9 |- F/_ x [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C
18	17	nfel1 2343	. . . . . . . 8 |- F/ x [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V
19		csbeq1a 3081	. . . . . . . . 9 |- ( x = ( 1st ` z ) -> [_ ( 2nd ` z ) / y ]_ C = [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C )
20	19	eleq1d 2258	. . . . . . . 8 |- ( x = ( 1st ` z ) -> ( [_ ( 2nd ` z ) / y ]_ C e. _V <-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V ) )
21	16, 18, 20	spcgf 2834	. . . . . . 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 1885	. . . 4 |- ( A. x A. y C e. V -> A. z [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V )
25	24	3ad2ant1 1020	. . 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 4249	. . . 4 |- ( ( R e. W /\ S e. X ) -> <. R , S >. e. _V )
27	26	3adant1 1017	. . 3 |- ( ( A. x A. y C e. V /\ R e. W /\ S e. X ) -> <. R , S >. e. _V )
28		fmpo.1	. . . . 5 |- F = ( x e. A , y e. B |-> C )
29		mpomptsx 6226	. . . . 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 2210	. . . 4 |- F = ( z e. U_ x e. A ( { x } X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C )
31	30	mptfvex 5625	. . 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 411	. 2 |- ( ( A. x A. y C e. V /\ R e. W /\ S e. X ) -> ( F ` <. R , S >. ) e. _V )
33	1, 32	eqeltrid 2276	1 |- ( ( A. x A. y C e. V /\ R e. W /\ S e. X ) -> ( R F S ) e. _V )

Syntax hints:   -> wi 4   /\ w3a 980   A.wal 1362   = wceq 1364   e. wcel 2160   _Vcvv 2752   [_csb 3072   {csn 3610   <.cop 3613   U_ciun 3904   |-> cmpt 4082   X. cxp 4645   ` cfv 5238  (class class class)co 5900   e. cmpo 5902   1stc1st 6167   2ndc2nd 6168

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-fo 5244  df-fv 5246  df-ov 5903  df-oprab 5904  df-mpo 5905  df-1st 6169  df-2nd 6170

This theorem is referenced by:  mpofvexi  6235  oaexg  6477  omexg  6480  oeiexg  6482  rhmex  13532