Theorem compsscnv 10354

Description: Complementation on a power set lattice is an involution. (Contributed by Mario Carneiro, 17-May-2015.)

Hypothesis

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

Assertion

Ref	Expression
compsscnv	⊢ ◡𝐹 = 𝐹

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem compsscnv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnvopab 6138	. 2 ⊢ ◡{⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))}
2		compss.a	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))
3		difeq2 4083	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑦))
4	3	cbvmptv 5219	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) = (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑦))
5		df-mpt 5197	. . . 4 ⊢ (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑦)) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))}
6	2, 4, 5	3eqtri 2796	. . 3 ⊢ 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))}
7	6	cnveqi 5861	. 2 ⊢ ◡𝐹 = ◡{⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))}
8		df-mpt 5197	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥))}
9		compsscnvlem 10353	. . . . 5 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦)) → (𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)))
10		compsscnvlem 10353	. . . . 5 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦)))
11	9, 10	impbii 212	. . . 4 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦)) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)))
12	11	opabbii 5182	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥))}
13	8, 2, 12	3eqtr4i 2802	. 2 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))}
14	1, 7, 13	3eqtr4i 2802	1 ⊢ ◡𝐹 = 𝐹

Syntax hints:   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ∖ cdif 3910  𝒫 cpw 4567  {copab 5177   ↦ cmpt 5196  ^◡ccnv 5661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-mpt 5197  df-xp 5668  df-rel 5669  df-cnv 5670

This theorem is referenced by:  compssiso  10357  isf34lem3  10358  compss  10359  isf34lem5  10361