Theorem fniniseg 5435

Description: Membership in the preimage of a singleton, under a function. (Contributed by Mario Carneiro, 12-May-2014.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
fniniseg	|- ( F Fn A -> ( C e. ( `' F " { B } ) <-> ( C e. A /\ ( F ` C ) = B ) ) )

Proof of Theorem fniniseg

Step	Hyp	Ref	Expression
1		elpreima 5434	. 2 |- ( F Fn A -> ( C e. ( `' F " { B } ) <-> ( C e. A /\ ( F ` C ) e. { B } ) ) )
2		funfvex 5337	. . . . 5 |- ( ( Fun F /\ C e. dom F ) -> ( F ` C ) e. _V )
3		elsng 3467	. . . . 5 |- ( ( F ` C ) e. _V -> ( ( F ` C ) e. { B } <-> ( F ` C ) = B ) )
4	2, 3	syl 14	. . . 4 |- ( ( Fun F /\ C e. dom F ) -> ( ( F ` C ) e. { B } <-> ( F ` C ) = B ) )
5	4	funfni 5129	. . 3 |- ( ( F Fn A /\ C e. A ) -> ( ( F ` C ) e. { B } <-> ( F ` C ) = B ) )
6	5	pm5.32da 441	. 2 |- ( F Fn A -> ( ( C e. A /\ ( F ` C ) e. { B } ) <-> ( C e. A /\ ( F ` C ) = B ) ) )
7	1, 6	bitrd 187	1 |- ( F Fn A -> ( C e. ( `' F " { B } ) <-> ( C e. A /\ ( F ` C ) = B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1290   e. wcel 1439   _Vcvv 2622   {csn 3452   `'ccnv 4453   dom cdm 4454   "cima 4457   Fun wfun 5024   Fn wfn 5025   ` cfv 5030

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 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-sbc 2844  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-id 4131  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-fv 5038

This theorem is referenced by: (None)