Theorem compsscnv 10368

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 6137	. 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 4115	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑦))
4	3	cbvmptv 5260	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) = (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑦))
5		df-mpt 5231	. . . 4 ⊢ (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑦)) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))}
6	2, 4, 5	3eqtri 2762	. . 3 ⊢ 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))}
7	6	cnveqi 5873	. 2 ⊢ ◡𝐹 = ◡{⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))}
8		df-mpt 5231	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥))}
9		compsscnvlem 10367	. . . . 5 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦)) → (𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)))
10		compsscnvlem 10367	. . . . 5 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦)))
11	9, 10	impbii 208	. . . 4 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦)) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)))
12	11	opabbii 5214	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥))}
13	8, 2, 12	3eqtr4i 2768	. 2 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))}
14	1, 7, 13	3eqtr4i 2768	1 ⊢ ◡𝐹 = 𝐹

Syntax hints:   ∧ wa 394   = wceq 1539   ∈ wcel 2104   ∖ cdif 3944  𝒫 cpw 4601  {copab 5209   ↦ cmpt 5230  ^◡ccnv 5674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-mpt 5231  df-xp 5681  df-rel 5682  df-cnv 5683

This theorem is referenced by:  compssiso  10371  isf34lem3  10372  compss  10373  isf34lem5  10375