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Theorem fniniseg 6336
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 6335 . 2 (𝐹 Fn 𝐴 → (𝐶 ∈ (𝐹 “ {𝐵}) ↔ (𝐶𝐴 ∧ (𝐹𝐶) ∈ {𝐵})))
2 fvex 6199 . . . 4 (𝐹𝐶) ∈ V
32elsn 4190 . . 3 ((𝐹𝐶) ∈ {𝐵} ↔ (𝐹𝐶) = 𝐵)
43anbi2i 730 . 2 ((𝐶𝐴 ∧ (𝐹𝐶) ∈ {𝐵}) ↔ (𝐶𝐴 ∧ (𝐹𝐶) = 𝐵))
51, 4syl6bb 276 1 (𝐹 Fn 𝐴 → (𝐶 ∈ (𝐹 “ {𝐵}) ↔ (𝐶𝐴 ∧ (𝐹𝐶) = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  {csn 4175  ccnv 5111  cima 5115   Fn wfn 5881  cfv 5886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-fv 5894
This theorem is referenced by:  fparlem1  7274  fparlem2  7275  pw2f1olem  8061  recmulnq  9783  dmrecnq  9787  vdwlem1  15679  vdwlem2  15680  vdwlem6  15684  vdwlem8  15686  vdwlem9  15687  vdwlem12  15690  vdwlem13  15691  ramval  15706  ramub1lem1  15724  ghmeqker  17681  efgrelexlemb  18157  efgredeu  18159  psgnevpmb  19927  qtopeu  21513  itg1addlem1  23453  i1faddlem  23454  i1fmullem  23455  i1fmulclem  23463  i1fres  23466  itg10a  23471  itg1ge0a  23472  itg1climres  23475  mbfi1fseqlem4  23479  ply1remlem  23916  ply1rem  23917  fta1glem1  23919  fta1glem2  23920  fta1g  23921  fta1blem  23922  plyco0  23942  ofmulrt  24031  plyremlem  24053  plyrem  24054  fta1lem  24056  fta1  24057  vieta1lem1  24059  vieta1lem2  24060  vieta1  24061  plyexmo  24062  elaa  24065  aannenlem1  24077  aalioulem2  24082  pilem1  24199  efif1olem3  24284  efif1olem4  24285  efifo  24287  eff1olem  24288  basellem4  24804  lgsqrlem2  25066  lgsqrlem3  25067  rpvmasum2  25195  dirith  25212  foresf1o  29327  ofpreima  29450  1stpreimas  29468  locfinreflem  29892  qqhre  30049  indpi1  30067  indpreima  30072  sibfof  30387  cvmliftlem6  31257  cvmliftlem7  31258  cvmliftlem8  31259  cvmliftlem9  31260  taupilem3  33145  itg2addnclem  33441  itg2addnclem2  33442  pw2f1o2val2  37433  dnnumch3  37443  proot1mul  37603  proot1hash  37604  proot1ex  37605  wessf1ornlem  39193
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