Theorem mpofvex 6357

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

Hypothesis

Ref	Expression
mpofvex.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   y, B

Allowed substitution hints:   B( x)   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 6010	. 2 |- ( R F S ) = ( F ` <. R , S >. )
2		elex 2811	. . . . . . . . 9 |- ( C e. V -> C e. _V )
3	2	alimi 1501	. . . . . . . 8 |- ( A. y C e. V -> A. y C e. _V )
4		vex 2802	. . . . . . . . 9 |- z e. _V
5		2ndexg 6320	. . . . . . . . 9 |- ( z e. _V -> ( 2nd ` z ) e. _V )
6		nfcv 2372	. . . . . . . . . 10 |- F/_ y ( 2nd ` z )
7		nfcsb1v 3157	. . . . . . . . . . 11 |- F/_ y [_ ( 2nd ` z ) / y ]_ C
8	7	nfel1 2383	. . . . . . . . . 10 |- F/ y [_ ( 2nd ` z ) / y ]_ C e. _V
9		csbeq1a 3133	. . . . . . . . . . 11 |- ( y = ( 2nd ` z ) -> C = [_ ( 2nd ` z ) / y ]_ C )
10	9	eleq1d 2298	. . . . . . . . . 10 |- ( y = ( 2nd ` z ) -> ( C e. _V <-> [_ ( 2nd ` z ) / y ]_ C e. _V ) )
11	6, 8, 10	spcgf 2885	. . . . . . . . 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 1501	. . . . . 6 |- ( A. x A. y C e. V -> A. x [_ ( 2nd ` z ) / y ]_ C e. _V )
15		1stexg 6319	. . . . . . 7 |- ( z e. _V -> ( 1st ` z ) e. _V )
16		nfcv 2372	. . . . . . . 8 |- F/_ x ( 1st ` z )
17		nfcsb1v 3157	. . . . . . . . 9 |- F/_ x [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C
18	17	nfel1 2383	. . . . . . . 8 |- F/ x [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V
19		csbeq1a 3133	. . . . . . . . 9 |- ( x = ( 1st ` z ) -> [_ ( 2nd ` z ) / y ]_ C = [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C )
20	19	eleq1d 2298	. . . . . . . 8 |- ( x = ( 1st ` z ) -> ( [_ ( 2nd ` z ) / y ]_ C e. _V <-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V ) )
21	16, 18, 20	spcgf 2885	. . . . . . 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 1920	. . . 4 |- ( A. x A. y C e. V -> A. z [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V )
25	24	3ad2ant1 1042	. . 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 4314	. . . 4 |- ( ( R e. W /\ S e. X ) -> <. R , S >. e. _V )
27	26	3adant1 1039	. . 3 |- ( ( A. x A. y C e. V /\ R e. W /\ S e. X ) -> <. R , S >. e. _V )
28		mpofvex.1	. . . . 5 |- F = ( x e. A , y e. B |-> C )
29		mpomptsx 6349	. . . . 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 2250	. . . 4 |- F = ( z e. U_ x e. A ( { x } X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C )
31	30	mptfvex 5722	. . 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 2316	1 |- ( ( A. x A. y C e. V /\ R e. W /\ S e. X ) -> ( R F S ) e. _V )

Syntax hints:   -> wi 4   /\ w3a 1002   A.wal 1393   = wceq 1395   e. wcel 2200   _Vcvv 2799   [_csb 3124   {csn 3666   <.cop 3669   U_ciun 3965   |-> cmpt 4145   X. cxp 4717   ` cfv 5318  (class class class)co 6007   e. cmpo 6009   1stc1st 6290   2ndc2nd 6291

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fo 5324  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293

This theorem is referenced by:  mpofvexi  6358  oaexg  6602  omexg  6605  oeiexg  6607  rhmex  14129