Theorem fniniseg 5580

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 5579	. 2 |- ( F Fn A -> ( C e. ( `' F " { B } ) <-> ( C e. A /\ ( F ` C ) e. { B } ) ) )
2		funfvex 5478	. . . . 5 |- ( ( Fun F /\ C e. dom F ) -> ( F ` C ) e. _V )
3		elsng 3571	. . . . 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 5263	. . 3 |- ( ( F Fn A /\ C e. A ) -> ( ( F ` C ) e. { B } <-> ( F ` C ) = B ) )
6	5	pm5.32da 448	. 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 1332   e. wcel 2125   _Vcvv 2709   {csn 3556   `'ccnv 4578   dom cdm 4579   "cima 4582   Fun wfun 5157   Fn wfn 5158   ` cfv 5163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-sbc 2934  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-opab 4022  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-fv 5171

This theorem is referenced by:  pilem1  13039  taupi  13582