Theorem compssiso 10380

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 5298	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝑥) ∈ V)
2	1	ralrimivw 3160	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 𝐴(𝐴 ∖ 𝑥) ∈ V)
3		compss.a	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))
4	3	fnmpt 6676	. . . 4 ⊢ (∀𝑥 ∈ 𝒫 𝐴(𝐴 ∖ 𝑥) ∈ V → 𝐹 Fn 𝒫 𝐴)
5	2, 4	syl 18	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐹 Fn 𝒫 𝐴)
6	3	compsscnv 10377	. . . . 5 ⊢ ◡𝐹 = 𝐹
7	6	fneq1i 6633	. . . 4 ⊢ (◡𝐹 Fn 𝒫 𝐴 ↔ 𝐹 Fn 𝒫 𝐴)
8	5, 7	sylibr 237	. . 3 ⊢ (𝐴 ∈ 𝑉 → ◡𝐹 Fn 𝒫 𝐴)
9		dff1o4 6830	. . 3 ⊢ (𝐹:𝒫 𝐴–1-1-onto→𝒫 𝐴 ↔ (𝐹 Fn 𝒫 𝐴 ∧ ◡𝐹 Fn 𝒫 𝐴))
10	5, 8, 9	sylanbrc 595	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐹:𝒫 𝐴–1-1-onto→𝒫 𝐴)
11		elpwi 4567	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝒫 𝐴 → 𝑏 ⊆ 𝐴)
12	11	ad2antll 742	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → 𝑏 ⊆ 𝐴)
13	3	isf34lem1 10378	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝐴) → (𝐹‘𝑏) = (𝐴 ∖ 𝑏))
14	12, 13	syldan 603	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝐹‘𝑏) = (𝐴 ∖ 𝑏))
15		elpwi 4567	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝐴)
16	15	ad2antrl 741	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → 𝑎 ⊆ 𝐴)
17	3	isf34lem1 10378	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑎 ⊆ 𝐴) → (𝐹‘𝑎) = (𝐴 ∖ 𝑎))
18	16, 17	syldan 603	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝐹‘𝑎) = (𝐴 ∖ 𝑎))
19	14, 18	psseq12d 4048	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → ((𝐹‘𝑏) ⊊ (𝐹‘𝑎) ↔ (𝐴 ∖ 𝑏) ⊊ (𝐴 ∖ 𝑎)))
20		difss 4086	. . . . . . 7 ⊢ (𝐴 ∖ 𝑎) ⊆ 𝐴
21		pssdifcom1 4448	. . . . . . 7 ⊢ ((𝑏 ⊆ 𝐴 ∧ (𝐴 ∖ 𝑎) ⊆ 𝐴) → ((𝐴 ∖ 𝑏) ⊊ (𝐴 ∖ 𝑎) ↔ (𝐴 ∖ (𝐴 ∖ 𝑎)) ⊊ 𝑏))
22	12, 20, 21	sylancl 598	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → ((𝐴 ∖ 𝑏) ⊊ (𝐴 ∖ 𝑎) ↔ (𝐴 ∖ (𝐴 ∖ 𝑎)) ⊊ 𝑏))
23		dfss4 4218	. . . . . . . 8 ⊢ (𝑎 ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ 𝑎)) = 𝑎)
24	16, 23	sylib 221	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝐴 ∖ (𝐴 ∖ 𝑎)) = 𝑎)
25	24	psseq1d 4046	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → ((𝐴 ∖ (𝐴 ∖ 𝑎)) ⊊ 𝑏 ↔ 𝑎 ⊊ 𝑏))
26	19, 22, 25	3bitrrd 309	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝑎 ⊊ 𝑏 ↔ (𝐹‘𝑏) ⊊ (𝐹‘𝑎)))
27		vex 3457	. . . . . 6 ⊢ 𝑏 ∈ V
28	27	brrpss 7731	. . . . 5 ⊢ (𝑎 [⊊] 𝑏 ↔ 𝑎 ⊊ 𝑏)
29		fvex 6895	. . . . . 6 ⊢ (𝐹‘𝑎) ∈ V
30	29	brrpss 7731	. . . . 5 ⊢ ((𝐹‘𝑏) [⊊] (𝐹‘𝑎) ↔ (𝐹‘𝑏) ⊊ (𝐹‘𝑎))
31	26, 28, 30	3bitr4g 317	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝑎 [⊊] 𝑏 ↔ (𝐹‘𝑏) [⊊] (𝐹‘𝑎)))
32		relrpss 7729	. . . . 5 ⊢ Rel [⊊]
33	32	relbrcnv 6107	. . . 4 ⊢ ((𝐹‘𝑎)◡ [⊊] (𝐹‘𝑏) ↔ (𝐹‘𝑏) [⊊] (𝐹‘𝑎))
34	31, 33	bitr4di 292	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝑎 [⊊] 𝑏 ↔ (𝐹‘𝑎)◡ [⊊] (𝐹‘𝑏)))
35	34	ralrimivva 3207	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝑎 [⊊] 𝑏 ↔ (𝐹‘𝑎)◡ [⊊] (𝐹‘𝑏)))
36		df-isom 6546	. 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 595	1 ⊢ (𝐴 ∈ 𝑉 → 𝐹 Isom [⊊] , ◡ [⊊] (𝒫 𝐴, 𝒫 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  Vcvv 3453   ∖ cdif 3899   ⊆ wss 3902   ⊊ wpss 3903  𝒫 cpw 4560   class class class wbr 5107   ↦ cmpt 5190  ^◡ccnv 5658   Fn wfn 6532  –1-1-onto→wf1o 6536  ‘cfv 6537   Isom wiso 6538   [⊊] crpss 7727

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-rpss 7728

This theorem is used by:  isf34lem3  10381  isf34lem5  10384  isfin1-4  10393