Theorem mptfvex 5424

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 2950	. . 3 |- ( y = C -> [_ y / x ]_ B = [_ C / x ]_ B )
2		fvmpt2.1	. . . 4 |- F = ( x e. A |-> B )
3		nfcv 2235	. . . . 5 |- F/_ y B
4		nfcsb1v 2977	. . . . 5 |- F/_ x [_ y / x ]_ B
5		csbeq1a 2955	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
6	3, 4, 5	cbvmpt 3955	. . . 4 |- ( x e. A |-> B ) = ( y e. A |-> [_ y / x ]_ B )
7	2, 6	eqtri 2115	. . 3 |- F = ( y e. A |-> [_ y / x ]_ B )
8	1, 7	fvmptss2 5414	. 2 |- ( F ` C ) C_ [_ C / x ]_ B
9		elex 2644	. . . . . 6 |- ( B e. V -> B e. _V )
10	9	alimi 1396	. . . . 5 |- ( A. x B e. V -> A. x B e. _V )
11	3	nfel1 2246	. . . . . 6 |- F/ y B e. _V
12	4	nfel1 2246	. . . . . 6 |- F/ x [_ y / x ]_ B e. _V
13	5	eleq1d 2163	. . . . . 6 |- ( x = y -> ( B e. _V <-> [_ y / x ]_ B e. _V ) )
14	11, 12, 13	cbval 1691	. . . . 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 2163	. . . . 5 |- ( y = C -> ( [_ y / x ]_ B e. _V <-> [_ C / x ]_ B e. _V ) )
17	16	spcgv 2720	. . . 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 3999	. 2 |- ( ( ( F ` C ) C_ [_ C / x ]_ B /\ [_ C / x ]_ B e. _V ) -> ( F ` C ) e. _V )
21	8, 19, 20	sylancr 406	1 |- ( ( A. x B e. V /\ C e. W ) -> ( F ` C ) e. _V )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1294   = wceq 1296   e. wcel 1445   _Vcvv 2633   [_csb 2947   C_ wss 3013   |-> cmpt 3921   ` cfv 5049

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-csb 2948  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-iota 5014  df-fun 5051  df-fv 5057

This theorem is referenced by:  mpt2fvex  6011  xpcomco  6622