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Theorem compsscnv 9781
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 5990 . 2 {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
2 compss.a . . . 4 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
3 difeq2 4090 . . . . 5 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
43cbvmptv 5160 . . . 4 (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) = (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴𝑦))
5 df-mpt 5138 . . . 4 (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴𝑦)) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
62, 4, 53eqtri 2845 . . 3 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
76cnveqi 5738 . 2 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
8 df-mpt 5138 . . 3 (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥))}
9 compsscnvlem 9780 . . . . 5 ((𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦)) → (𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)))
10 compsscnvlem 9780 . . . . 5 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦)))
119, 10impbii 210 . . . 4 ((𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦)) ↔ (𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)))
1211opabbii 5124 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥))}
138, 2, 123eqtr4i 2851 . 2 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦))}
141, 7, 133eqtr4i 2851 1 𝐹 = 𝐹
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  wcel 2105  cdif 3930  𝒫 cpw 4535  {copab 5119  cmpt 5137  ccnv 5547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-mpt 5138  df-xp 5554  df-rel 5555  df-cnv 5556
This theorem is referenced by:  compssiso  9784  isf34lem3  9785  compss  9786  isf34lem5  9788
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