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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 5452 | . 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 3684 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑦)) | |
4 | 3 | cbvmptv 4678 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) = (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑦)) |
5 | df-mpt 4645 | . . . 4 ⊢ (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑦)) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))} | |
6 | 2, 4, 5 | 3eqtri 2636 | . . 3 ⊢ 𝐹 = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))} |
7 | 6 | cnveqi 5219 | . 2 ⊢ ◡𝐹 = ◡{〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))} |
8 | df-mpt 4645 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥))} | |
9 | compsscnvlem 9075 | . . . . 5 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦)) → (𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥))) | |
10 | compsscnvlem 9075 | . . . . 5 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))) | |
11 | 9, 10 | impbii 198 | . . . 4 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦)) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥))) |
12 | 11 | opabbii 4649 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥))} |
13 | 8, 2, 12 | 3eqtr4i 2642 | . 2 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))} |
14 | 1, 7, 13 | 3eqtr4i 2642 | 1 ⊢ ◡𝐹 = 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 𝒫 cpw 4108 {copab 4642 ↦ cmpt 4643 ◡ccnv 5037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-mpt 4645 df-xp 5044 df-rel 5045 df-cnv 5046 |
This theorem is referenced by: compssiso 9079 isf34lem3 9080 compss 9081 isf34lem5 9083 |
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