Theorem fniniseg 5797

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 5796	. 2 |- ( F Fn A -> ( C e. ( `' F " { B } ) <-> ( C e. A /\ ( F ` C ) e. { B } ) ) )
2		funfvex 5686	. . . . 5 |- ( ( Fun F /\ C e. dom F ) -> ( F ` C ) e. _V )
3		elsng 3703	. . . . 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 5457	. . 3 |- ( ( F Fn A /\ C e. A ) -> ( ( F ` C ) e. { B } <-> ( F ` C ) = B ) )
6	5	pm5.32da 452	. 2 |- ( F Fn A -> ( ( C e. A /\ ( F ` C ) e. { B } ) <-> ( C e. A /\ ( F ` C ) = B ) ) )
7	1, 6	bitrd 188	1 |- ( F Fn A -> ( C e. ( `' F " { B } ) <-> ( C e. A /\ ( F ` C ) = B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   _Vcvv 2812   {csn 3688   `'ccnv 4747   dom cdm 4748   "cima 4751   Fun wfun 5345   Fn wfn 5346   ` cfv 5351

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-sbc 3042  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-fv 5359

This theorem is referenced by:  pw2f1odclem  7086  ghmeqker  13977  pilem1  15631  taupi  16845