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Theorem fniniseg 5315
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 5314 . 2 (𝐹 Fn 𝐴 → (𝐶 ∈ (𝐹 “ {𝐵}) ↔ (𝐶𝐴 ∧ (𝐹𝐶) ∈ {𝐵})))
2 funfvex 5220 . . . . 5 ((Fun 𝐹𝐶 ∈ dom 𝐹) → (𝐹𝐶) ∈ V)
3 elsng 3418 . . . . 5 ((𝐹𝐶) ∈ V → ((𝐹𝐶) ∈ {𝐵} ↔ (𝐹𝐶) = 𝐵))
42, 3syl 14 . . . 4 ((Fun 𝐹𝐶 ∈ dom 𝐹) → ((𝐹𝐶) ∈ {𝐵} ↔ (𝐹𝐶) = 𝐵))
54funfni 5027 . . 3 ((𝐹 Fn 𝐴𝐶𝐴) → ((𝐹𝐶) ∈ {𝐵} ↔ (𝐹𝐶) = 𝐵))
65pm5.32da 433 . 2 (𝐹 Fn 𝐴 → ((𝐶𝐴 ∧ (𝐹𝐶) ∈ {𝐵}) ↔ (𝐶𝐴 ∧ (𝐹𝐶) = 𝐵)))
71, 6bitrd 181 1 (𝐹 Fn 𝐴 → (𝐶 ∈ (𝐹 “ {𝐵}) ↔ (𝐶𝐴 ∧ (𝐹𝐶) = 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  Vcvv 2574  {csn 3403  ccnv 4372  dom cdm 4373  cima 4376  Fun wfun 4924   Fn wfn 4925  cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-fv 4938
This theorem is referenced by: (None)
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