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Mirrors > Home > MPE Home > Th. List > compsscnv | Structured version Visualization version GIF version |
Description: Complementation on a power set lattice is an involution. (Contributed by Mario Carneiro, 17-May-2015.) |
Ref | Expression |
---|---|
compss.a | ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) |
Ref | Expression |
---|---|
compsscnv | ⊢ ◡𝐹 = 𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvopab 6139 | . 2 ⊢ ◡{⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))} | |
2 | compss.a | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) | |
3 | difeq2 4117 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑦)) | |
4 | 3 | cbvmptv 5262 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) = (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑦)) |
5 | df-mpt 5233 | . . . 4 ⊢ (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑦)) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))} | |
6 | 2, 4, 5 | 3eqtri 2765 | . . 3 ⊢ 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))} |
7 | 6 | cnveqi 5875 | . 2 ⊢ ◡𝐹 = ◡{⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))} |
8 | df-mpt 5233 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥))} | |
9 | compsscnvlem 10365 | . . . . 5 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦)) → (𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥))) | |
10 | compsscnvlem 10365 | . . . . 5 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))) | |
11 | 9, 10 | impbii 208 | . . . 4 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦)) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥))) |
12 | 11 | opabbii 5216 | . . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥))} |
13 | 8, 2, 12 | 3eqtr4i 2771 | . 2 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))} |
14 | 1, 7, 13 | 3eqtr4i 2771 | 1 ⊢ ◡𝐹 = 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∖ cdif 3946 𝒫 cpw 4603 {copab 5211 ↦ cmpt 5232 ◡ccnv 5676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-mpt 5233 df-xp 5683 df-rel 5684 df-cnv 5685 |
This theorem is referenced by: compssiso 10369 isf34lem3 10370 compss 10371 isf34lem5 10373 |
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