Theorem mptfvex 5499

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

Hypothesis

Ref	Expression
fvmpt2.1	|- F = ( x e. A |-> B )

Assertion

Ref	Expression
mptfvex	|- ( ( A. x B e. V /\ C e. W ) -> ( F ` C ) e. _V )

Distinct variable groups:   x, A   x, C

Allowed substitution hints:   B( x)   F( x)   V( x)   W( x)

Proof of Theorem mptfvex

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3001	. . 3 |- ( y = C -> [_ y / x ]_ B = [_ C / x ]_ B )
2		fvmpt2.1	. . . 4 |- F = ( x e. A |-> B )
3		nfcv 2279	. . . . 5 |- F/_ y B
4		nfcsb1v 3030	. . . . 5 |- F/_ x [_ y / x ]_ B
5		csbeq1a 3007	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
6	3, 4, 5	cbvmpt 4018	. . . 4 |- ( x e. A |-> B ) = ( y e. A |-> [_ y / x ]_ B )
7	2, 6	eqtri 2158	. . 3 |- F = ( y e. A |-> [_ y / x ]_ B )
8	1, 7	fvmptss2 5489	. 2 |- ( F ` C ) C_ [_ C / x ]_ B
9		elex 2692	. . . . . 6 |- ( B e. V -> B e. _V )
10	9	alimi 1431	. . . . 5 |- ( A. x B e. V -> A. x B e. _V )
11	3	nfel1 2290	. . . . . 6 |- F/ y B e. _V
12	4	nfel1 2290	. . . . . 6 |- F/ x [_ y / x ]_ B e. _V
13	5	eleq1d 2206	. . . . . 6 |- ( x = y -> ( B e. _V <-> [_ y / x ]_ B e. _V ) )
14	11, 12, 13	cbval 1727	. . . . 5 |- ( A. x B e. _V <-> A. y [_ y / x ]_ B e. _V )
15	10, 14	sylib 121	. . . 4 |- ( A. x B e. V -> A. y [_ y / x ]_ B e. _V )
16	1	eleq1d 2206	. . . . 5 |- ( y = C -> ( [_ y / x ]_ B e. _V <-> [_ C / x ]_ B e. _V ) )
17	16	spcgv 2768	. . . 4 |- ( C e. W -> ( A. y [_ y / x ]_ B e. _V -> [_ C / x ]_ B e. _V ) )
18	15, 17	syl5 32	. . 3 |- ( C e. W -> ( A. x B e. V -> [_ C / x ]_ B e. _V ) )
19	18	impcom 124	. 2 |- ( ( A. x B e. V /\ C e. W ) -> [_ C / x ]_ B e. _V )
20		ssexg 4062	. 2 |- ( ( ( F ` C ) C_ [_ C / x ]_ B /\ [_ C / x ]_ B e. _V ) -> ( F ` C ) e. _V )
21	8, 19, 20	sylancr 410	1 |- ( ( A. x B e. V /\ C e. W ) -> ( F ` C ) e. _V )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1329   = wceq 1331   e. wcel 1480   _Vcvv 2681   [_csb 2998   C_ wss 3066   |-> cmpt 3984   ` cfv 5118

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-iota 5083  df-fun 5120  df-fv 5126

This theorem is referenced by:  mpofvex  6094  xpcomco  6713