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Theorem compsscnv 9137
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 5492 . 2 {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
2 compss.a . . . 4 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
3 difeq2 3700 . . . . 5 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
43cbvmptv 4710 . . . 4 (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) = (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴𝑦))
5 df-mpt 4675 . . . 4 (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴𝑦)) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
62, 4, 53eqtri 2647 . . 3 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
76cnveqi 5257 . 2 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
8 df-mpt 4675 . . 3 (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥))}
9 compsscnvlem 9136 . . . . 5 ((𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦)) → (𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)))
10 compsscnvlem 9136 . . . . 5 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦)))
119, 10impbii 199 . . . 4 ((𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦)) ↔ (𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)))
1211opabbii 4679 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥))}
138, 2, 123eqtr4i 2653 . 2 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
141, 7, 133eqtr4i 2653 1 𝐹 = 𝐹
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1987  cdif 3552  𝒫 cpw 4130  {copab 4672  cmpt 4673  ccnv 5073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-mpt 4675  df-xp 5080  df-rel 5081  df-cnv 5082
This theorem is referenced by:  compssiso  9140  isf34lem3  9141  compss  9142  isf34lem5  9144
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