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Theorem elfi 8865
Description: Specific properties of an element of (fi‘𝐵). (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
Assertion
Ref Expression
elfi ((𝐴𝑉𝐵𝑊) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = 𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑉   𝑥,𝑊

Proof of Theorem elfi
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fival 8864 . . 3 (𝐵𝑊 → (fi‘𝐵) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑦 = 𝑥})
21eleq2d 2878 . 2 (𝐵𝑊 → (𝐴 ∈ (fi‘𝐵) ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑦 = 𝑥}))
3 eqeq1 2805 . . . 4 (𝑦 = 𝐴 → (𝑦 = 𝑥𝐴 = 𝑥))
43rexbidv 3259 . . 3 (𝑦 = 𝐴 → (∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑦 = 𝑥 ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = 𝑥))
54elabg 3617 . 2 (𝐴𝑉 → (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑦 = 𝑥} ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = 𝑥))
62, 5sylan9bbr 514 1 ((𝐴𝑉𝐵𝑊) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  {cab 2779  wrex 3110  cin 3883  𝒫 cpw 4500   cint 4841  cfv 6328  Fincfn 8496  ficfi 8862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-fi 8863
This theorem is referenced by:  elfi2  8866  elfir  8867  inelfi  8870  fiin  8874  dffi2  8875  elfiun  8882  subbascn  21862  cmpfi  22016  fbasfip  22476  alexsubALTlem4  22658  zarcmplem  31234  heibor1lem  35240  elrfi  39622
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