Theorem isf34lem3 9791

Description: Lemma for isfin3-4 9798. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)

Hypothesis

Ref	Expression
compss.a	⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))

Assertion

Ref	Expression
isf34lem3	⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ 𝑋)) = 𝑋)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

Allowed substitution hints:   𝐹(𝑥)   𝑋(𝑥)

Proof of Theorem isf34lem3

Step	Hyp	Ref	Expression
1		compss.a	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))
2	1	compsscnv 9787	. . 3 ⊢ ◡𝐹 = 𝐹
3	2	imaeq1i 5920	. 2 ⊢ (◡𝐹 “ (𝐹 “ 𝑋)) = (𝐹 “ (𝐹 “ 𝑋))
4	1	compssiso 9790	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐹 Isom [⊊] , ◡ [⊊] (𝒫 𝐴, 𝒫 𝐴))
5		isof1o 7070	. . . 4 ⊢ (𝐹 Isom [⊊] , ◡ [⊊] (𝒫 𝐴, 𝒫 𝐴) → 𝐹:𝒫 𝐴–1-1-onto→𝒫 𝐴)
6		f1of1 6608	. . . 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 6625	. . 3 ⊢ ((𝐹:𝒫 𝐴–1-1→𝒫 𝐴 ∧ 𝑋 ⊆ 𝒫 𝐴) → (◡𝐹 “ (𝐹 “ 𝑋)) = 𝑋)
9	7, 8	sylan 582	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝒫 𝐴) → (◡𝐹 “ (𝐹 “ 𝑋)) = 𝑋)
10	3, 9	syl5eqr 2870	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ 𝑋)) = 𝑋)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ∖ cdif 3932   ⊆ wss 3935  𝒫 cpw 4538   ↦ cmpt 5138  ^◡ccnv 5548   “ cima 5552  –1-1→wf1 6346  –1-1-onto→wf1o 6348   Isom wiso 6350   [⊊] crpss 7442

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-rpss 7443

This theorem is referenced by:  isf34lem5  9794  isf34lem7  9795  isf34lem6  9796