Theorem mptfvex 5732

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 3130	. . 3 |- ( y = C -> [_ y / x ]_ B = [_ C / x ]_ B )
2		fvmpt2.1	. . . 4 |- F = ( x e. A |-> B )
3		nfcv 2374	. . . . 5 |- F/_ y B
4		nfcsb1v 3160	. . . . 5 |- F/_ x [_ y / x ]_ B
5		csbeq1a 3136	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
6	3, 4, 5	cbvmpt 4184	. . . 4 |- ( x e. A |-> B ) = ( y e. A |-> [_ y / x ]_ B )
7	2, 6	eqtri 2252	. . 3 |- F = ( y e. A |-> [_ y / x ]_ B )
8	1, 7	fvmptss2 5721	. 2 |- ( F ` C ) C_ [_ C / x ]_ B
9		elex 2814	. . . . . 6 |- ( B e. V -> B e. _V )
10	9	alimi 1503	. . . . 5 |- ( A. x B e. V -> A. x B e. _V )
11	3	nfel1 2385	. . . . . 6 |- F/ y B e. _V
12	4	nfel1 2385	. . . . . 6 |- F/ x [_ y / x ]_ B e. _V
13	5	eleq1d 2300	. . . . . 6 |- ( x = y -> ( B e. _V <-> [_ y / x ]_ B e. _V ) )
14	11, 12, 13	cbval 1802	. . . . 5 |- ( A. x B e. _V <-> A. y [_ y / x ]_ B e. _V )
15	10, 14	sylib 122	. . . 4 |- ( A. x B e. V -> A. y [_ y / x ]_ B e. _V )
16	1	eleq1d 2300	. . . . 5 |- ( y = C -> ( [_ y / x ]_ B e. _V <-> [_ C / x ]_ B e. _V ) )
17	16	spcgv 2893	. . . 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 125	. 2 |- ( ( A. x B e. V /\ C e. W ) -> [_ C / x ]_ B e. _V )
20		ssexg 4228	. 2 |- ( ( ( F ` C ) C_ [_ C / x ]_ B /\ [_ C / x ]_ B e. _V ) -> ( F ` C ) e. _V )
21	8, 19, 20	sylancr 414	1 |- ( ( A. x B e. V /\ C e. W ) -> ( F ` C ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1395   = wceq 1397   e. wcel 2202   _Vcvv 2802   [_csb 3127   C_ wss 3200   |-> cmpt 4150   ` cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-iota 5286  df-fun 5328  df-fv 5334

This theorem is referenced by:  mpofvex  6369  xpcomco  7009  lssex  14367  mopnset  14565  metuex  14568