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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnvtrrel | Structured version Visualization version GIF version |
Description: The converse of a transitive relation is a transitive relation. (Contributed by RP, 25-Dec-2019.) |
Ref | Expression |
---|---|
cnvtrrel | ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ (◡𝑆 ∘ ◡𝑆) ⊆ ◡𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvss 5881 | . . 3 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 → ◡(𝑆 ∘ 𝑆) ⊆ ◡𝑆) | |
2 | cnvss 5881 | . . . 4 ⊢ (◡(𝑆 ∘ 𝑆) ⊆ ◡𝑆 → ◡◡(𝑆 ∘ 𝑆) ⊆ ◡◡𝑆) | |
3 | cnvco 5894 | . . . . . . . . 9 ⊢ ◡(𝑆 ∘ 𝑆) = (◡𝑆 ∘ ◡𝑆) | |
4 | 3 | cnveqi 5883 | . . . . . . . 8 ⊢ ◡◡(𝑆 ∘ 𝑆) = ◡(◡𝑆 ∘ ◡𝑆) |
5 | cnvco 5894 | . . . . . . . 8 ⊢ ◡(◡𝑆 ∘ ◡𝑆) = (◡◡𝑆 ∘ ◡◡𝑆) | |
6 | cocnvcnv1 6275 | . . . . . . . . 9 ⊢ (◡◡𝑆 ∘ ◡◡𝑆) = (𝑆 ∘ ◡◡𝑆) | |
7 | cocnvcnv2 6276 | . . . . . . . . 9 ⊢ (𝑆 ∘ ◡◡𝑆) = (𝑆 ∘ 𝑆) | |
8 | 6, 7 | eqtri 2764 | . . . . . . . 8 ⊢ (◡◡𝑆 ∘ ◡◡𝑆) = (𝑆 ∘ 𝑆) |
9 | 4, 5, 8 | 3eqtri 2768 | . . . . . . 7 ⊢ ◡◡(𝑆 ∘ 𝑆) = (𝑆 ∘ 𝑆) |
10 | 9 | sseq1i 4011 | . . . . . 6 ⊢ (◡◡(𝑆 ∘ 𝑆) ⊆ ◡◡𝑆 ↔ (𝑆 ∘ 𝑆) ⊆ ◡◡𝑆) |
11 | 10 | biimpi 216 | . . . . 5 ⊢ (◡◡(𝑆 ∘ 𝑆) ⊆ ◡◡𝑆 → (𝑆 ∘ 𝑆) ⊆ ◡◡𝑆) |
12 | cnvcnvss 6212 | . . . . 5 ⊢ ◡◡𝑆 ⊆ 𝑆 | |
13 | 11, 12 | sstrdi 3995 | . . . 4 ⊢ (◡◡(𝑆 ∘ 𝑆) ⊆ ◡◡𝑆 → (𝑆 ∘ 𝑆) ⊆ 𝑆) |
14 | 2, 13 | syl 17 | . . 3 ⊢ (◡(𝑆 ∘ 𝑆) ⊆ ◡𝑆 → (𝑆 ∘ 𝑆) ⊆ 𝑆) |
15 | 1, 14 | impbii 209 | . 2 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ ◡(𝑆 ∘ 𝑆) ⊆ ◡𝑆) |
16 | 3 | sseq1i 4011 | . 2 ⊢ (◡(𝑆 ∘ 𝑆) ⊆ ◡𝑆 ↔ (◡𝑆 ∘ ◡𝑆) ⊆ ◡𝑆) |
17 | 15, 16 | bitri 275 | 1 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ (◡𝑆 ∘ ◡𝑆) ⊆ ◡𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ⊆ wss 3950 ◡ccnv 5682 ∘ ccom 5687 |
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-ext 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430 |
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 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5142 df-opab 5204 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 |
This theorem is referenced by: (None) |
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