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Theorem compsscnv 9782
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 5995 . 2 {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
2 compss.a . . . 4 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
3 difeq2 4097 . . . . 5 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
43cbvmptv 5166 . . . 4 (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) = (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴𝑦))
5 df-mpt 5144 . . . 4 (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴𝑦)) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
62, 4, 53eqtri 2853 . . 3 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
76cnveqi 5744 . 2 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
8 df-mpt 5144 . . 3 (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥))}
9 compsscnvlem 9781 . . . . 5 ((𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦)) → (𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)))
10 compsscnvlem 9781 . . . . 5 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦)))
119, 10impbii 210 . . . 4 ((𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦)) ↔ (𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)))
1211opabbii 5130 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥))}
138, 2, 123eqtr4i 2859 . 2 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
141, 7, 133eqtr4i 2859 1 𝐹 = 𝐹
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1530  wcel 2107  cdif 3937  𝒫 cpw 4542  {copab 5125  cmpt 5143  ccnv 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-mpt 5144  df-xp 5560  df-rel 5561  df-cnv 5562
This theorem is referenced by:  compssiso  9785  isf34lem3  9786  compss  9787  isf34lem5  9789
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