Theorem mpofvex 6163

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 5839	. 2 |- ( R F S ) = ( F ` <. R , S >. )
2		elex 2732	. . . . . . . . 9 |- ( C e. V -> C e. _V )
3	2	alimi 1442	. . . . . . . 8 |- ( A. y C e. V -> A. y C e. _V )
4		vex 2724	. . . . . . . . 9 |- z e. _V
5		2ndexg 6128	. . . . . . . . 9 |- ( z e. _V -> ( 2nd ` z ) e. _V )
6		nfcv 2306	. . . . . . . . . 10 |- F/_ y ( 2nd ` z )
7		nfcsb1v 3073	. . . . . . . . . . 11 |- F/_ y [_ ( 2nd ` z ) / y ]_ C
8	7	nfel1 2317	. . . . . . . . . 10 |- F/ y [_ ( 2nd ` z ) / y ]_ C e. _V
9		csbeq1a 3049	. . . . . . . . . . 11 |- ( y = ( 2nd ` z ) -> C = [_ ( 2nd ` z ) / y ]_ C )
10	9	eleq1d 2233	. . . . . . . . . 10 |- ( y = ( 2nd ` z ) -> ( C e. _V <-> [_ ( 2nd ` z ) / y ]_ C e. _V ) )
11	6, 8, 10	spcgf 2803	. . . . . . . . 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 1442	. . . . . 6 |- ( A. x A. y C e. V -> A. x [_ ( 2nd ` z ) / y ]_ C e. _V )
15		1stexg 6127	. . . . . . 7 |- ( z e. _V -> ( 1st ` z ) e. _V )
16		nfcv 2306	. . . . . . . 8 |- F/_ x ( 1st ` z )
17		nfcsb1v 3073	. . . . . . . . 9 |- F/_ x [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C
18	17	nfel1 2317	. . . . . . . 8 |- F/ x [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V
19		csbeq1a 3049	. . . . . . . . 9 |- ( x = ( 1st ` z ) -> [_ ( 2nd ` z ) / y ]_ C = [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C )
20	19	eleq1d 2233	. . . . . . . 8 |- ( x = ( 1st ` z ) -> ( [_ ( 2nd ` z ) / y ]_ C e. _V <-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V ) )
21	16, 18, 20	spcgf 2803	. . . . . . 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 1861	. . . 4 |- ( A. x A. y C e. V -> A. z [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V )
25	24	3ad2ant1 1007	. . 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 4200	. . . 4 |- ( ( R e. W /\ S e. X ) -> <. R , S >. e. _V )
27	26	3adant1 1004	. . 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 6157	. . . . 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 2185	. . . 4 |- F = ( z e. U_ x e. A ( { x } X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C )
31	30	mptfvex 5565	. . 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 409	. 2 |- ( ( A. x A. y C e. V /\ R e. W /\ S e. X ) -> ( F ` <. R , S >. ) e. _V )
33	1, 32	eqeltrid 2251	1 |- ( ( A. x A. y C e. V /\ R e. W /\ S e. X ) -> ( R F S ) e. _V )

Syntax hints:   -> wi 4   /\ w3a 967   A.wal 1340   = wceq 1342   e. wcel 2135   _Vcvv 2721   [_csb 3040   {csn 3570   <.cop 3573   U_ciun 3860   |-> cmpt 4037   X. cxp 4596   ` cfv 5182  (class class class)co 5836   e. cmpo 5838   1stc1st 6098   2ndc2nd 6099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-sbc 2947  df-csb 3041  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-fo 5188  df-fv 5190  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101

This theorem is referenced by:  mpofvexi  6166  oaexg  6407  omexg  6410  oeiexg  6412