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Theorem elfi 8263
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 8262 . . 3 (𝐵𝑊 → (fi‘𝐵) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑦 = 𝑥})
21eleq2d 2684 . 2 (𝐵𝑊 → (𝐴 ∈ (fi‘𝐵) ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑦 = 𝑥}))
3 eqeq1 2625 . . . 4 (𝑦 = 𝐴 → (𝑦 = 𝑥𝐴 = 𝑥))
43rexbidv 3045 . . 3 (𝑦 = 𝐴 → (∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑦 = 𝑥 ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = 𝑥))
54elabg 3334 . 2 (𝐴𝑉 → (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑦 = 𝑥} ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = 𝑥))
62, 5sylan9bbr 736 1 ((𝐴𝑉𝐵𝑊) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {cab 2607  wrex 2908  cin 3554  𝒫 cpw 4130   cint 4440  cfv 5847  Fincfn 7899  ficfi 8260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-fi 8261
This theorem is referenced by:  elfi2  8264  elfir  8265  inelfi  8268  fiin  8272  dffi2  8273  elfiun  8280  subbascn  20968  cmpfi  21121  fbasfip  21582  alexsubALTlem4  21764  heibor1lem  33237  elrfi  36734
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