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| Description: Specific properties of an element of (fi‘𝐵). (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.) | 
| Ref | Expression | 
|---|---|
| elfi | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = ∩ 𝑥)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fival 9452 | . . 3 ⊢ (𝐵 ∈ 𝑊 → (fi‘𝐵) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑦 = ∩ 𝑥}) | |
| 2 | 1 | eleq2d 2827 | . 2 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ∈ (fi‘𝐵) ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑦 = ∩ 𝑥})) | 
| 3 | eqeq1 2741 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 = ∩ 𝑥 ↔ 𝐴 = ∩ 𝑥)) | |
| 4 | 3 | rexbidv 3179 | . . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑦 = ∩ 𝑥 ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = ∩ 𝑥)) | 
| 5 | 4 | elabg 3676 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑦 = ∩ 𝑥} ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = ∩ 𝑥)) | 
| 6 | 2, 5 | sylan9bbr 510 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = ∩ 𝑥)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 ∃wrex 3070 ∩ cin 3950 𝒫 cpw 4600 ∩ cint 4946 ‘cfv 6561 Fincfn 8985 ficfi 9450 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-fi 9451 | 
| This theorem is referenced by: elfi2 9454 elfir 9455 inelfi 9458 fiin 9462 dffi2 9463 elfiun 9470 subbascn 23262 cmpfi 23416 fbasfip 23876 alexsubALTlem4 24058 zarcmplem 33880 heibor1lem 37816 elrfi 42705 | 
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