![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 8733 | . 2 ⊢ (𝑅 Er 𝐴 → Rel 𝑅) | |
2 | relcnv 6108 | . . 3 ⊢ Rel ◡𝑅 | |
3 | id 22 | . . . . . 6 ⊢ (𝑅 Er 𝐴 → 𝑅 Er 𝐴) | |
4 | 3 | ersymb 8738 | . . . . 5 ⊢ (𝑅 Er 𝐴 → (𝑦𝑅𝑥 ↔ 𝑥𝑅𝑦)) |
5 | vex 3475 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
6 | vex 3475 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
7 | 5, 6 | brcnv 5885 | . . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) |
8 | df-br 5149 | . . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅) | |
9 | 7, 8 | bitr3i 277 | . . . . 5 ⊢ (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅) |
10 | df-br 5149 | . . . . 5 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅) | |
11 | 4, 9, 10 | 3bitr3g 313 | . . . 4 ⊢ (𝑅 Er 𝐴 → (⟨𝑥, 𝑦⟩ ∈ ◡𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)) |
12 | 11 | eqrelrdv2 5797 | . . 3 ⊢ (((Rel ◡𝑅 ∧ Rel 𝑅) ∧ 𝑅 Er 𝐴) → ◡𝑅 = 𝑅) |
13 | 2, 12 | mpanl1 699 | . 2 ⊢ ((Rel 𝑅 ∧ 𝑅 Er 𝐴) → ◡𝑅 = 𝑅) |
14 | 1, 13 | mpancom 687 | 1 ⊢ (𝑅 Er 𝐴 → ◡𝑅 = 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ⟨cop 4635 class class class wbr 5148 ◡ccnv 5677 Rel wrel 5683 Er wer 8721 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-xp 5684 df-rel 5685 df-cnv 5686 df-er 8724 |
This theorem is referenced by: errn 8746 prjspeclsp 42036 |
Copyright terms: Public domain | W3C validator |