Theorem compssiso 10284

Description: Complementation is an antiautomorphism on power set lattices. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)

Hypothesis

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

Assertion

Ref	Expression
compssiso	⊢ (𝐴 ∈ 𝑉 → 𝐹 Isom [⊊] , ◡ [⊊] (𝒫 𝐴, 𝒫 𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem compssiso

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		difexg 5274	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝑥) ∈ V)
2	1	ralrimivw 3132	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 𝐴(𝐴 ∖ 𝑥) ∈ V)
3		compss.a	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))
4	3	fnmpt 6632	. . . 4 ⊢ (∀𝑥 ∈ 𝒫 𝐴(𝐴 ∖ 𝑥) ∈ V → 𝐹 Fn 𝒫 𝐴)
5	2, 4	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐹 Fn 𝒫 𝐴)
6	3	compsscnv 10281	. . . . 5 ⊢ ◡𝐹 = 𝐹
7	6	fneq1i 6589	. . . 4 ⊢ (◡𝐹 Fn 𝒫 𝐴 ↔ 𝐹 Fn 𝒫 𝐴)
8	5, 7	sylibr 234	. . 3 ⊢ (𝐴 ∈ 𝑉 → ◡𝐹 Fn 𝒫 𝐴)
9		dff1o4 6782	. . 3 ⊢ (𝐹:𝒫 𝐴–1-1-onto→𝒫 𝐴 ↔ (𝐹 Fn 𝒫 𝐴 ∧ ◡𝐹 Fn 𝒫 𝐴))
10	5, 8, 9	sylanbrc 583	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐹:𝒫 𝐴–1-1-onto→𝒫 𝐴)
11		elpwi 4561	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝒫 𝐴 → 𝑏 ⊆ 𝐴)
12	11	ad2antll 729	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → 𝑏 ⊆ 𝐴)
13	3	isf34lem1 10282	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝐴) → (𝐹‘𝑏) = (𝐴 ∖ 𝑏))
14	12, 13	syldan 591	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝐹‘𝑏) = (𝐴 ∖ 𝑏))
15		elpwi 4561	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝐴)
16	15	ad2antrl 728	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → 𝑎 ⊆ 𝐴)
17	3	isf34lem1 10282	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑎 ⊆ 𝐴) → (𝐹‘𝑎) = (𝐴 ∖ 𝑎))
18	16, 17	syldan 591	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝐹‘𝑎) = (𝐴 ∖ 𝑎))
19	14, 18	psseq12d 4049	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → ((𝐹‘𝑏) ⊊ (𝐹‘𝑎) ↔ (𝐴 ∖ 𝑏) ⊊ (𝐴 ∖ 𝑎)))
20		difss 4088	. . . . . . 7 ⊢ (𝐴 ∖ 𝑎) ⊆ 𝐴
21		pssdifcom1 4442	. . . . . . 7 ⊢ ((𝑏 ⊆ 𝐴 ∧ (𝐴 ∖ 𝑎) ⊆ 𝐴) → ((𝐴 ∖ 𝑏) ⊊ (𝐴 ∖ 𝑎) ↔ (𝐴 ∖ (𝐴 ∖ 𝑎)) ⊊ 𝑏))
22	12, 20, 21	sylancl 586	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → ((𝐴 ∖ 𝑏) ⊊ (𝐴 ∖ 𝑎) ↔ (𝐴 ∖ (𝐴 ∖ 𝑎)) ⊊ 𝑏))
23		dfss4 4221	. . . . . . . 8 ⊢ (𝑎 ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ 𝑎)) = 𝑎)
24	16, 23	sylib 218	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝐴 ∖ (𝐴 ∖ 𝑎)) = 𝑎)
25	24	psseq1d 4047	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → ((𝐴 ∖ (𝐴 ∖ 𝑎)) ⊊ 𝑏 ↔ 𝑎 ⊊ 𝑏))
26	19, 22, 25	3bitrrd 306	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝑎 ⊊ 𝑏 ↔ (𝐹‘𝑏) ⊊ (𝐹‘𝑎)))
27		vex 3444	. . . . . 6 ⊢ 𝑏 ∈ V
28	27	brrpss 7671	. . . . 5 ⊢ (𝑎 [⊊] 𝑏 ↔ 𝑎 ⊊ 𝑏)
29		fvex 6847	. . . . . 6 ⊢ (𝐹‘𝑎) ∈ V
30	29	brrpss 7671	. . . . 5 ⊢ ((𝐹‘𝑏) [⊊] (𝐹‘𝑎) ↔ (𝐹‘𝑏) ⊊ (𝐹‘𝑎))
31	26, 28, 30	3bitr4g 314	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝑎 [⊊] 𝑏 ↔ (𝐹‘𝑏) [⊊] (𝐹‘𝑎)))
32		relrpss 7669	. . . . 5 ⊢ Rel [⊊]
33	32	relbrcnv 6066	. . . 4 ⊢ ((𝐹‘𝑎)◡ [⊊] (𝐹‘𝑏) ↔ (𝐹‘𝑏) [⊊] (𝐹‘𝑎))
34	31, 33	bitr4di 289	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝑎 [⊊] 𝑏 ↔ (𝐹‘𝑎)◡ [⊊] (𝐹‘𝑏)))
35	34	ralrimivva 3179	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝑎 [⊊] 𝑏 ↔ (𝐹‘𝑎)◡ [⊊] (𝐹‘𝑏)))
36		df-isom 6501	. 2 ⊢ (𝐹 Isom [⊊] , ◡ [⊊] (𝒫 𝐴, 𝒫 𝐴) ↔ (𝐹:𝒫 𝐴–1-1-onto→𝒫 𝐴 ∧ ∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝑎 [⊊] 𝑏 ↔ (𝐹‘𝑎)◡ [⊊] (𝐹‘𝑏))))
37	10, 35, 36	sylanbrc 583	1 ⊢ (𝐴 ∈ 𝑉 → 𝐹 Isom [⊊] , ◡ [⊊] (𝒫 𝐴, 𝒫 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  Vcvv 3440   ∖ cdif 3898   ⊆ wss 3901   ⊊ wpss 3902  𝒫 cpw 4554   class class class wbr 5098   ↦ cmpt 5179  ^◡ccnv 5623   Fn wfn 6487  –1-1-onto→wf1o 6491  ‘cfv 6492   Isom wiso 6493   [⊊] crpss 7667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-rpss 7668

This theorem is referenced by:  isf34lem3  10285  isf34lem5  10288  isfin1-4  10297