Theorem fniniseg 6000

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 5999	. 2 |- ( F Fn A -> ( C e. ( `' F " { B } ) <-> ( C e. A /\ ( F ` C ) e. { B } ) ) )
2		fvex 5874	. . . 4 |- ( F ` C ) e. _V
3	2	elsnc 4051	. . 3 |- ( ( F ` C ) e. { B } <-> ( F ` C ) = B )
4	3	anbi2i 694	. 2 |- ( ( C e. A /\ ( F ` C ) e. { B } ) <-> ( C e. A /\ ( F ` C ) = B ) )
5	1, 4	syl6bb 261	1 |- ( F Fn A -> ( C e. ( `' F " { B } ) <-> ( C e. A /\ ( F ` C ) = B ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   {csn 4027   `'ccnv 4998   "cima 5002   Fn wfn 5581   ` cfv 5586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594

This theorem is referenced by:  fparlem1  6880  fparlem2  6881  pw2f1olem  7618  recmulnq  9338  dmrecnq  9342  vdwlem1  14354  vdwlem2  14355  vdwlem6  14359  vdwlem8  14361  vdwlem9  14362  vdwlem12  14365  vdwlem13  14366  ramval  14381  ramub1lem1  14399  ghmeqker  16088  efgrelexlemb  16564  efgredeu  16566  psgnevpmb  18390  qtopeu  19952  itg1addlem1  21834  i1faddlem  21835  i1fmullem  21836  i1fmulclem  21844  i1fres  21847  itg10a  21852  itg1ge0a  21853  itg1climres  21856  mbfi1fseqlem4  21860  ply1remlem  22298  ply1rem  22299  fta1glem1  22301  fta1glem2  22302  fta1g  22303  fta1blem  22304  plyco0  22324  ofmulrt  22412  plyremlem  22434  plyrem  22435  fta1lem  22437  fta1  22438  vieta1lem1  22440  vieta1lem2  22441  vieta1  22442  plyexmo  22443  elaa  22446  aannenlem1  22458  aalioulem2  22463  pilem1  22580  efif1olem3  22664  efif1olem4  22665  efifo  22667  eff1olem  22668  basellem4  23085  lgsqrlem2  23345  lgsqrlem3  23346  rpvmasum2  23425  dirith  23442  ofpreima  27179  qqhre  27634  indpi1  27675  indpreima  27678  sibfof  27922  cvmliftlem6  28375  cvmliftlem7  28376  cvmliftlem8  28377  cvmliftlem9  28378  itg2addnclem  29643  itg2addnclem2  29644  pw2f1o2val2  30586  dnnumch3  30597  proot1mul  30761  proot1hash  30765  proot1ex  30766  taupilem3  36764