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Theorem elfi 8479
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 8478 . . 3 (𝐵𝑊 → (fi‘𝐵) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑦 = 𝑥})
21eleq2d 2836 . 2 (𝐵𝑊 → (𝐴 ∈ (fi‘𝐵) ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑦 = 𝑥}))
3 eqeq1 2775 . . . 4 (𝑦 = 𝐴 → (𝑦 = 𝑥𝐴 = 𝑥))
43rexbidv 3200 . . 3 (𝑦 = 𝐴 → (∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑦 = 𝑥 ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = 𝑥))
54elabg 3502 . 2 (𝐴𝑉 → (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑦 = 𝑥} ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = 𝑥))
62, 5sylan9bbr 500 1 ((𝐴𝑉𝐵𝑊) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  {cab 2757  wrex 3062  cin 3722  𝒫 cpw 4298   cint 4612  cfv 6030  Fincfn 8113  ficfi 8476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-int 4613  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-fi 8477
This theorem is referenced by:  elfi2  8480  elfir  8481  inelfi  8484  fiin  8488  dffi2  8489  elfiun  8496  subbascn  21279  cmpfi  21432  fbasfip  21892  alexsubALTlem4  22074  heibor1lem  33940  elrfi  37783
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