Theorem compsscnv 10456

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 6133	. 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 4056	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑦))
4	3	cbvmptv 5208	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) = (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑦))
5		df-mpt 5186	. . . 4 ⊢ (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑦)) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))}
6	2, 4, 5	3eqtri 2787	. . 3 ⊢ 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))}
7	6	cnveqi 5853	. 2 ⊢ ◡𝐹 = ◡{⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))}
8		df-mpt 5186	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥))}
9		compsscnvlem 10455	. . . . 5 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦)) → (𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)))
10		compsscnvlem 10455	. . . . 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 5171	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥))}
13	8, 2, 12	3eqtr4i 2793	. 2 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))}
14	1, 7, 13	3eqtr4i 2793	1 ⊢ ◡𝐹 = 𝐹

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ∖ cdif 4044  𝒫 cpw 4556  {copab 5166   ↦ cmpt 5185  ^◡ccnv 5650

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-11 2193  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-mpt 5186  df-xp 5657  df-rel 5658  df-cnv 5659

This theorem is used by:  compssiso  10459  isf34lem3  10460  compss  10461  isf34lem5  10463