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Theorem compsscnv 9076
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 ↦ (𝐴𝑥))
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.
StepHypRef Expression
1 cnvopab 5452 . 2 {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
2 compss.a . . . 4 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
3 difeq2 3684 . . . . 5 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
43cbvmptv 4678 . . . 4 (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) = (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴𝑦))
5 df-mpt 4645 . . . 4 (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴𝑦)) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
62, 4, 53eqtri 2636 . . 3 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
76cnveqi 5219 . 2 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
8 df-mpt 4645 . . 3 (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥))}
9 compsscnvlem 9075 . . . . 5 ((𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦)) → (𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)))
10 compsscnvlem 9075 . . . . 5 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦)))
119, 10impbii 198 . . . 4 ((𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦)) ↔ (𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)))
1211opabbii 4649 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥))}
138, 2, 123eqtr4i 2642 . 2 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
141, 7, 133eqtr4i 2642 1 𝐹 = 𝐹
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1475  wcel 1977  cdif 3537  𝒫 cpw 4108  {copab 4642  cmpt 4643  ccnv 5037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-mpt 4645  df-xp 5044  df-rel 5045  df-cnv 5046
This theorem is referenced by:  compssiso  9079  isf34lem3  9080  compss  9081  isf34lem5  9083
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