Theorem fniniseg 5639

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 5638	. 2 |- ( F Fn A -> ( C e. ( `' F " { B } ) <-> ( C e. A /\ ( F ` C ) e. { B } ) ) )
2		funfvex 5534	. . . . 5 |- ( ( Fun F /\ C e. dom F ) -> ( F ` C ) e. _V )
3		elsng 3609	. . . . 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 5318	. . 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 1353   e. wcel 2148   _Vcvv 2739   {csn 3594   `'ccnv 4627   dom cdm 4628   "cima 4631   Fun wfun 5212   Fn wfn 5213   ` cfv 5218

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226

This theorem is referenced by:  pilem1  14340  taupi  14962