Theorem compssiso 10371

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 5327	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝑥) ∈ V)
2	1	ralrimivw 3150	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 𝐴(𝐴 ∖ 𝑥) ∈ V)
3		compss.a	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))
4	3	fnmpt 6690	. . . 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 10368	. . . . 5 ⊢ ◡𝐹 = 𝐹
7	6	fneq1i 6646	. . . 4 ⊢ (◡𝐹 Fn 𝒫 𝐴 ↔ 𝐹 Fn 𝒫 𝐴)
8	5, 7	sylibr 233	. . 3 ⊢ (𝐴 ∈ 𝑉 → ◡𝐹 Fn 𝒫 𝐴)
9		dff1o4 6841	. . 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 4609	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝒫 𝐴 → 𝑏 ⊆ 𝐴)
12	11	ad2antll 727	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → 𝑏 ⊆ 𝐴)
13	3	isf34lem1 10369	. . . . . . . 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 4609	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝐴)
16	15	ad2antrl 726	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → 𝑎 ⊆ 𝐴)
17	3	isf34lem1 10369	. . . . . . . 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 4094	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → ((𝐹‘𝑏) ⊊ (𝐹‘𝑎) ↔ (𝐴 ∖ 𝑏) ⊊ (𝐴 ∖ 𝑎)))
20		difss 4131	. . . . . . 7 ⊢ (𝐴 ∖ 𝑎) ⊆ 𝐴
21		pssdifcom1 4489	. . . . . . 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 4258	. . . . . . . 8 ⊢ (𝑎 ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ 𝑎)) = 𝑎)
24	16, 23	sylib 217	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝐴 ∖ (𝐴 ∖ 𝑎)) = 𝑎)
25	24	psseq1d 4092	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → ((𝐴 ∖ (𝐴 ∖ 𝑎)) ⊊ 𝑏 ↔ 𝑎 ⊊ 𝑏))
26	19, 22, 25	3bitrrd 305	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝑎 ⊊ 𝑏 ↔ (𝐹‘𝑏) ⊊ (𝐹‘𝑎)))
27		vex 3478	. . . . . 6 ⊢ 𝑏 ∈ V
28	27	brrpss 7718	. . . . 5 ⊢ (𝑎 [⊊] 𝑏 ↔ 𝑎 ⊊ 𝑏)
29		fvex 6904	. . . . . 6 ⊢ (𝐹‘𝑎) ∈ V
30	29	brrpss 7718	. . . . 5 ⊢ ((𝐹‘𝑏) [⊊] (𝐹‘𝑎) ↔ (𝐹‘𝑏) ⊊ (𝐹‘𝑎))
31	26, 28, 30	3bitr4g 313	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝑎 [⊊] 𝑏 ↔ (𝐹‘𝑏) [⊊] (𝐹‘𝑎)))
32		relrpss 7716	. . . . 5 ⊢ Rel [⊊]
33	32	relbrcnv 6106	. . . 4 ⊢ ((𝐹‘𝑎)◡ [⊊] (𝐹‘𝑏) ↔ (𝐹‘𝑏) [⊊] (𝐹‘𝑎))
34	31, 33	bitr4di 288	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝑎 [⊊] 𝑏 ↔ (𝐹‘𝑎)◡ [⊊] (𝐹‘𝑏)))
35	34	ralrimivva 3200	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝑎 [⊊] 𝑏 ↔ (𝐹‘𝑎)◡ [⊊] (𝐹‘𝑏)))
36		df-isom 6552	. 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 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  Vcvv 3474   ∖ cdif 3945   ⊆ wss 3948   ⊊ wpss 3949  𝒫 cpw 4602   class class class wbr 5148   ↦ cmpt 5231  ^◡ccnv 5675   Fn wfn 6538  –1-1-onto→wf1o 6542  ‘cfv 6543   Isom wiso 6544   [⊊] crpss 7714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-rpss 7715

This theorem is referenced by:  isf34lem3  10372  isf34lem5  10375  isfin1-4  10384