Theorem mptfvex 5713

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 3127	. . 3 |- ( y = C -> [_ y / x ]_ B = [_ C / x ]_ B )
2		fvmpt2.1	. . . 4 |- F = ( x e. A |-> B )
3		nfcv 2372	. . . . 5 |- F/_ y B
4		nfcsb1v 3157	. . . . 5 |- F/_ x [_ y / x ]_ B
5		csbeq1a 3133	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
6	3, 4, 5	cbvmpt 4178	. . . 4 |- ( x e. A |-> B ) = ( y e. A |-> [_ y / x ]_ B )
7	2, 6	eqtri 2250	. . 3 |- F = ( y e. A |-> [_ y / x ]_ B )
8	1, 7	fvmptss2 5702	. 2 |- ( F ` C ) C_ [_ C / x ]_ B
9		elex 2811	. . . . . 6 |- ( B e. V -> B e. _V )
10	9	alimi 1501	. . . . 5 |- ( A. x B e. V -> A. x B e. _V )
11	3	nfel1 2383	. . . . . 6 |- F/ y B e. _V
12	4	nfel1 2383	. . . . . 6 |- F/ x [_ y / x ]_ B e. _V
13	5	eleq1d 2298	. . . . . 6 |- ( x = y -> ( B e. _V <-> [_ y / x ]_ B e. _V ) )
14	11, 12, 13	cbval 1800	. . . . 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 2298	. . . . 5 |- ( y = C -> ( [_ y / x ]_ B e. _V <-> [_ C / x ]_ B e. _V ) )
17	16	spcgv 2890	. . . 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 4222	. 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 1393   = wceq 1395   e. wcel 2200   _Vcvv 2799   [_csb 3124   C_ wss 3197   |-> cmpt 4144   ` cfv 5314

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-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

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 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-iota 5274  df-fun 5316  df-fv 5322

This theorem is referenced by:  mpofvex  6341  xpcomco  6973  lssex  14303  mopnset  14501  metuex  14504