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Mirrors > Home > MPE Home > Th. List > ercnv | Structured version Visualization version GIF version |
Description: The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
ercnv | ⊢ (𝑅 Er 𝐴 → ◡𝑅 = 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | errel 8711 | . 2 ⊢ (𝑅 Er 𝐴 → Rel 𝑅) | |
2 | relcnv 6103 | . . 3 ⊢ Rel ◡𝑅 | |
3 | id 22 | . . . . . 6 ⊢ (𝑅 Er 𝐴 → 𝑅 Er 𝐴) | |
4 | 3 | ersymb 8716 | . . . . 5 ⊢ (𝑅 Er 𝐴 → (𝑦𝑅𝑥 ↔ 𝑥𝑅𝑦)) |
5 | vex 3478 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
6 | vex 3478 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
7 | 5, 6 | brcnv 5882 | . . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) |
8 | df-br 5149 | . . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅) | |
9 | 7, 8 | bitr3i 276 | . . . . 5 ⊢ (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅) |
10 | df-br 5149 | . . . . 5 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅) | |
11 | 4, 9, 10 | 3bitr3g 312 | . . . 4 ⊢ (𝑅 Er 𝐴 → (⟨𝑥, 𝑦⟩ ∈ ◡𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)) |
12 | 11 | eqrelrdv2 5795 | . . 3 ⊢ (((Rel ◡𝑅 ∧ Rel 𝑅) ∧ 𝑅 Er 𝐴) → ◡𝑅 = 𝑅) |
13 | 2, 12 | mpanl1 698 | . 2 ⊢ ((Rel 𝑅 ∧ 𝑅 Er 𝐴) → ◡𝑅 = 𝑅) |
14 | 1, 13 | mpancom 686 | 1 ⊢ (𝑅 Er 𝐴 → ◡𝑅 = 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ⟨cop 4634 class class class wbr 5148 ◡ccnv 5675 Rel wrel 5681 Er wer 8699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-cnv 5684 df-er 8702 |
This theorem is referenced by: errn 8724 prjspeclsp 41355 |
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