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Theorem compsscnv 10408
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 6159 . 2 {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
2 compss.a . . . 4 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
3 difeq2 4129 . . . . 5 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
43cbvmptv 5260 . . . 4 (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) = (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴𝑦))
5 df-mpt 5231 . . . 4 (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴𝑦)) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
62, 4, 53eqtri 2766 . . 3 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
76cnveqi 5887 . 2 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
8 df-mpt 5231 . . 3 (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥))}
9 compsscnvlem 10407 . . . . 5 ((𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦)) → (𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)))
10 compsscnvlem 10407 . . . . 5 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦)))
119, 10impbii 209 . . . 4 ((𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦)) ↔ (𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)))
1211opabbii 5214 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥))}
138, 2, 123eqtr4i 2772 . 2 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
141, 7, 133eqtr4i 2772 1 𝐹 = 𝐹
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1536  wcel 2105  cdif 3959  𝒫 cpw 4604  {copab 5209  cmpt 5230  ccnv 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-11 2154  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-mpt 5231  df-xp 5694  df-rel 5695  df-cnv 5696
This theorem is referenced by:  compssiso  10411  isf34lem3  10412  compss  10413  isf34lem5  10415
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