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Theorem fniniseg 5287
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 5286 . 2 (𝐹 Fn 𝐴 → (𝐶 ∈ (𝐹 “ {𝐵}) ↔ (𝐶𝐴 ∧ (𝐹𝐶) ∈ {𝐵})))
2 funfvex 5192 . . . . 5 ((Fun 𝐹𝐶 ∈ dom 𝐹) → (𝐹𝐶) ∈ V)
3 elsng 3390 . . . . 5 ((𝐹𝐶) ∈ V → ((𝐹𝐶) ∈ {𝐵} ↔ (𝐹𝐶) = 𝐵))
42, 3syl 14 . . . 4 ((Fun 𝐹𝐶 ∈ dom 𝐹) → ((𝐹𝐶) ∈ {𝐵} ↔ (𝐹𝐶) = 𝐵))
54funfni 4999 . . 3 ((𝐹 Fn 𝐴𝐶𝐴) → ((𝐹𝐶) ∈ {𝐵} ↔ (𝐹𝐶) = 𝐵))
65pm5.32da 425 . 2 (𝐹 Fn 𝐴 → ((𝐶𝐴 ∧ (𝐹𝐶) ∈ {𝐵}) ↔ (𝐶𝐴 ∧ (𝐹𝐶) = 𝐵)))
71, 6bitrd 177 1 (𝐹 Fn 𝐴 → (𝐶 ∈ (𝐹 “ {𝐵}) ↔ (𝐶𝐴 ∧ (𝐹𝐶) = 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wcel 1393  Vcvv 2557  {csn 3375  ccnv 4344  dom cdm 4345  cima 4348  Fun wfun 4896   Fn wfn 4897  cfv 4902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910
This theorem is referenced by: (None)
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