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Theorem elfi 9451
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 9450 . . 3 (𝐵𝑊 → (fi‘𝐵) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑦 = 𝑥})
21eleq2d 2825 . 2 (𝐵𝑊 → (𝐴 ∈ (fi‘𝐵) ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑦 = 𝑥}))
3 eqeq1 2739 . . . 4 (𝑦 = 𝐴 → (𝑦 = 𝑥𝐴 = 𝑥))
43rexbidv 3177 . . 3 (𝑦 = 𝐴 → (∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑦 = 𝑥 ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = 𝑥))
54elabg 3677 . 2 (𝐴𝑉 → (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑦 = 𝑥} ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = 𝑥))
62, 5sylan9bbr 510 1 ((𝐴𝑉𝐵𝑊) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {cab 2712  wrex 3068  cin 3962  𝒫 cpw 4605   cint 4951  cfv 6563  Fincfn 8984  ficfi 9448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-fi 9449
This theorem is referenced by:  elfi2  9452  elfir  9453  inelfi  9456  fiin  9460  dffi2  9461  elfiun  9468  subbascn  23278  cmpfi  23432  fbasfip  23892  alexsubALTlem4  24074  zarcmplem  33842  heibor1lem  37796  elrfi  42682
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