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Theorem fniniseg 5230
 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}) ↔ (𝐶 A (𝐹𝐶) = B)))

Proof of Theorem fniniseg
StepHypRef Expression
1 elpreima 5229 . 2 (𝐹 Fn A → (𝐶 (𝐹 “ {B}) ↔ (𝐶 A (𝐹𝐶) {B})))
2 funfvex 5135 . . . . 5 ((Fun 𝐹 𝐶 dom 𝐹) → (𝐹𝐶) V)
3 elsncg 3389 . . . . 5 ((𝐹𝐶) V → ((𝐹𝐶) {B} ↔ (𝐹𝐶) = B))
42, 3syl 14 . . . 4 ((Fun 𝐹 𝐶 dom 𝐹) → ((𝐹𝐶) {B} ↔ (𝐹𝐶) = B))
54funfni 4942 . . 3 ((𝐹 Fn A 𝐶 A) → ((𝐹𝐶) {B} ↔ (𝐹𝐶) = B))
65pm5.32da 425 . 2 (𝐹 Fn A → ((𝐶 A (𝐹𝐶) {B}) ↔ (𝐶 A (𝐹𝐶) = B)))
71, 6bitrd 177 1 (𝐹 Fn A → (𝐶 (𝐹 “ {B}) ↔ (𝐶 A (𝐹𝐶) = B)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  Vcvv 2551  {csn 3367  ◡ccnv 4287  dom cdm 4288   “ cima 4291  Fun wfun 4839   Fn wfn 4840  ‘cfv 4845 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853 This theorem is referenced by: (None)
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