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Mirrors > Home > MPE Home > Th. List > isf34lem3 | Structured version Visualization version GIF version |
Description: Lemma for isfin3-4 9087. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
Ref | Expression |
---|---|
compss.a | ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) |
Ref | Expression |
---|---|
isf34lem3 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ 𝑋)) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | compss.a | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) | |
2 | 1 | compsscnv 9076 | . . 3 ⊢ ◡𝐹 = 𝐹 |
3 | 2 | imaeq1i 5382 | . 2 ⊢ (◡𝐹 “ (𝐹 “ 𝑋)) = (𝐹 “ (𝐹 “ 𝑋)) |
4 | 1 | compssiso 9079 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐹 Isom [⊊] , ◡ [⊊] (𝒫 𝐴, 𝒫 𝐴)) |
5 | isof1o 6473 | . . . 4 ⊢ (𝐹 Isom [⊊] , ◡ [⊊] (𝒫 𝐴, 𝒫 𝐴) → 𝐹:𝒫 𝐴–1-1-onto→𝒫 𝐴) | |
6 | f1of1 6049 | . . . 4 ⊢ (𝐹:𝒫 𝐴–1-1-onto→𝒫 𝐴 → 𝐹:𝒫 𝐴–1-1→𝒫 𝐴) | |
7 | 4, 5, 6 | 3syl 18 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐹:𝒫 𝐴–1-1→𝒫 𝐴) |
8 | f1imacnv 6066 | . . 3 ⊢ ((𝐹:𝒫 𝐴–1-1→𝒫 𝐴 ∧ 𝑋 ⊆ 𝒫 𝐴) → (◡𝐹 “ (𝐹 “ 𝑋)) = 𝑋) | |
9 | 7, 8 | sylan 487 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝒫 𝐴) → (◡𝐹 “ (𝐹 “ 𝑋)) = 𝑋) |
10 | 3, 9 | syl5eqr 2658 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ 𝑋)) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ⊆ wss 3540 𝒫 cpw 4108 ↦ cmpt 4643 ◡ccnv 5037 “ cima 5041 –1-1→wf1 5801 –1-1-onto→wf1o 5803 Isom wiso 5805 [⊊] crpss 6834 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-rpss 6835 |
This theorem is referenced by: isf34lem5 9083 isf34lem7 9084 isf34lem6 9085 |
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