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Theorem fniniseg 6246
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 ∈ (𝐹 “ {𝐵}) ↔ (𝐶𝐴 ∧ (𝐹𝐶) = 𝐵)))

Proof of Theorem fniniseg
StepHypRef Expression
1 elpreima 6245 . 2 (𝐹 Fn 𝐴 → (𝐶 ∈ (𝐹 “ {𝐵}) ↔ (𝐶𝐴 ∧ (𝐹𝐶) ∈ {𝐵})))
2 fvex 6113 . . . 4 (𝐹𝐶) ∈ V
32elsn 4140 . . 3 ((𝐹𝐶) ∈ {𝐵} ↔ (𝐹𝐶) = 𝐵)
43anbi2i 726 . 2 ((𝐶𝐴 ∧ (𝐹𝐶) ∈ {𝐵}) ↔ (𝐶𝐴 ∧ (𝐹𝐶) = 𝐵))
51, 4syl6bb 275 1 (𝐹 Fn 𝐴 → (𝐶 ∈ (𝐹 “ {𝐵}) ↔ (𝐶𝐴 ∧ (𝐹𝐶) = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  {csn 4125  ccnv 5037  cima 5041   Fn wfn 5799  cfv 5804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812
This theorem is referenced by:  fparlem1  7164  fparlem2  7165  pw2f1olem  7949  recmulnq  9665  dmrecnq  9669  vdwlem1  15523  vdwlem2  15524  vdwlem6  15528  vdwlem8  15530  vdwlem9  15531  vdwlem12  15534  vdwlem13  15535  ramval  15550  ramub1lem1  15568  ghmeqker  17510  efgrelexlemb  17986  efgredeu  17988  psgnevpmb  19752  qtopeu  21329  itg1addlem1  23265  i1faddlem  23266  i1fmullem  23267  i1fmulclem  23275  i1fres  23278  itg10a  23283  itg1ge0a  23284  itg1climres  23287  mbfi1fseqlem4  23291  ply1remlem  23726  ply1rem  23727  fta1glem1  23729  fta1glem2  23730  fta1g  23731  fta1blem  23732  plyco0  23752  ofmulrt  23841  plyremlem  23863  plyrem  23864  fta1lem  23866  fta1  23867  vieta1lem1  23869  vieta1lem2  23870  vieta1  23871  plyexmo  23872  elaa  23875  aannenlem1  23887  aalioulem2  23892  pilem1  24009  efif1olem3  24094  efif1olem4  24095  efifo  24097  eff1olem  24098  basellem4  24610  lgsqrlem2  24872  lgsqrlem3  24873  rpvmasum2  25001  dirith  25018  foresf1o  28727  ofpreima  28848  1stpreimas  28866  locfinreflem  29235  qqhre  29392  indpi1  29411  indpreima  29414  sibfof  29729  cvmliftlem6  30526  cvmliftlem7  30527  cvmliftlem8  30528  cvmliftlem9  30529  taupilem3  32342  itg2addnclem  32631  itg2addnclem2  32632  pw2f1o2val2  36625  dnnumch3  36635  proot1mul  36796  proot1hash  36797  proot1ex  36798  wessf1ornlem  38366
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