Theorem mptfvex 5614

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 3072	. . 3 |- ( y = C -> [_ y / x ]_ B = [_ C / x ]_ B )
2		fvmpt2.1	. . . 4 |- F = ( x e. A |-> B )
3		nfcv 2329	. . . . 5 |- F/_ y B
4		nfcsb1v 3102	. . . . 5 |- F/_ x [_ y / x ]_ B
5		csbeq1a 3078	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
6	3, 4, 5	cbvmpt 4110	. . . 4 |- ( x e. A |-> B ) = ( y e. A |-> [_ y / x ]_ B )
7	2, 6	eqtri 2208	. . 3 |- F = ( y e. A |-> [_ y / x ]_ B )
8	1, 7	fvmptss2 5604	. 2 |- ( F ` C ) C_ [_ C / x ]_ B
9		elex 2760	. . . . . 6 |- ( B e. V -> B e. _V )
10	9	alimi 1465	. . . . 5 |- ( A. x B e. V -> A. x B e. _V )
11	3	nfel1 2340	. . . . . 6 |- F/ y B e. _V
12	4	nfel1 2340	. . . . . 6 |- F/ x [_ y / x ]_ B e. _V
13	5	eleq1d 2256	. . . . . 6 |- ( x = y -> ( B e. _V <-> [_ y / x ]_ B e. _V ) )
14	11, 12, 13	cbval 1764	. . . . 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 2256	. . . . 5 |- ( y = C -> ( [_ y / x ]_ B e. _V <-> [_ C / x ]_ B e. _V ) )
17	16	spcgv 2836	. . . 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 4154	. 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 1361   = wceq 1363   e. wcel 2158   _Vcvv 2749   [_csb 3069   C_ wss 3141   |-> cmpt 4076   ` cfv 5228

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-iota 5190  df-fun 5230  df-fv 5236

This theorem is referenced by:  mpofvex  6217  xpcomco  6839  lssex  13506