Theorem mpofvex 6431

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 6078	. 2 |- ( R F S ) = ( F ` <. R , S >. )
2		elex 2833	. . . . . . . . 9 |- ( C e. V -> C e. _V )
3	2	alimi 1508	. . . . . . . 8 |- ( A. y C e. V -> A. y C e. _V )
4		vex 2824	. . . . . . . . 9 |- z e. _V
5		2ndexg 6392	. . . . . . . . 9 |- ( z e. _V -> ( 2nd ` z ) e. _V )
6		nfcv 2392	. . . . . . . . . 10 |- F/_ y ( 2nd ` z )
7		nfcsb1v 3180	. . . . . . . . . . 11 |- F/_ y [_ ( 2nd ` z ) / y ]_ C
8	7	nfel1 2403	. . . . . . . . . 10 |- F/ y [_ ( 2nd ` z ) / y ]_ C e. _V
9		csbeq1a 3156	. . . . . . . . . . 11 |- ( y = ( 2nd ` z ) -> C = [_ ( 2nd ` z ) / y ]_ C )
10	9	eleq1d 2307	. . . . . . . . . 10 |- ( y = ( 2nd ` z ) -> ( C e. _V <-> [_ ( 2nd ` z ) / y ]_ C e. _V ) )
11	6, 8, 10	spcgf 2907	. . . . . . . . 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 1508	. . . . . 6 |- ( A. x A. y C e. V -> A. x [_ ( 2nd ` z ) / y ]_ C e. _V )
15		1stexg 6391	. . . . . . 7 |- ( z e. _V -> ( 1st ` z ) e. _V )
16		nfcv 2392	. . . . . . . 8 |- F/_ x ( 1st ` z )
17		nfcsb1v 3180	. . . . . . . . 9 |- F/_ x [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C
18	17	nfel1 2403	. . . . . . . 8 |- F/ x [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V
19		csbeq1a 3156	. . . . . . . . 9 |- ( x = ( 1st ` z ) -> [_ ( 2nd ` z ) / y ]_ C = [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C )
20	19	eleq1d 2307	. . . . . . . 8 |- ( x = ( 1st ` z ) -> ( [_ ( 2nd ` z ) / y ]_ C e. _V <-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V ) )
21	16, 18, 20	spcgf 2907	. . . . . . 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 1927	. . . 4 |- ( A. x A. y C e. V -> A. z [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V )
25	24	3ad2ant1 1049	. . 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 4363	. . . 4 |- ( ( R e. W /\ S e. X ) -> <. R , S >. e. _V )
27	26	3adant1 1046	. . 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 6423	. . . . 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 2259	. . . 4 |- F = ( z e. U_ x e. A ( { x } X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C )
31	30	mptfvex 5785	. . 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 415	. 2 |- ( ( A. x A. y C e. V /\ R e. W /\ S e. X ) -> ( F ` <. R , S >. ) e. _V )
33	1, 32	eqeltrid 2325	1 |- ( ( A. x A. y C e. V /\ R e. W /\ S e. X ) -> ( R F S ) e. _V )

Syntax hints:   -> wi 4   /\ w3a 1009   A.wal 1400   = wceq 1402   e. wcel 2209   _Vcvv 2821   [_csb 3147   {csn 3705   <.cop 3708   U_ciun 4007   |-> cmpt 4187   X. cxp 4767   ` cfv 5372  (class class class)co 6075   e. cmpo 6077   1stc1st 6362   2ndc2nd 6363

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fo 5378  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365

This theorem is referenced by:  mpofvexi  6432  oaexg  6711  omexg  6714  oeiexg  6716  rhmex  14437  clwwlknon  16584