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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 6157 | . 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 4120 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑦)) | |
| 4 | 3 | cbvmptv 5255 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) = (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑦)) |
| 5 | df-mpt 5226 | . . . 4 ⊢ (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑦)) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))} | |
| 6 | 2, 4, 5 | 3eqtri 2769 | . . 3 ⊢ 𝐹 = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))} |
| 7 | 6 | cnveqi 5885 | . 2 ⊢ ◡𝐹 = ◡{〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))} |
| 8 | df-mpt 5226 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥))} | |
| 9 | compsscnvlem 10410 | . . . . 5 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦)) → (𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥))) | |
| 10 | compsscnvlem 10410 | . . . . 5 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))) | |
| 11 | 9, 10 | impbii 209 | . . . 4 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦)) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥))) |
| 12 | 11 | opabbii 5210 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥))} |
| 13 | 8, 2, 12 | 3eqtr4i 2775 | . 2 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))} |
| 14 | 1, 7, 13 | 3eqtr4i 2775 | 1 ⊢ ◡𝐹 = 𝐹 |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 𝒫 cpw 4600 {copab 5205 ↦ cmpt 5225 ◡ccnv 5684 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-xp 5691 df-rel 5692 df-cnv 5693 |
| This theorem is referenced by: compssiso 10414 isf34lem3 10415 compss 10416 isf34lem5 10418 |
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