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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 8098 | . 2 ⊢ (𝑅 Er 𝐴 → Rel 𝑅) | |
2 | relcnv 5807 | . . 3 ⊢ Rel ◡𝑅 | |
3 | id 22 | . . . . . 6 ⊢ (𝑅 Er 𝐴 → 𝑅 Er 𝐴) | |
4 | 3 | ersymb 8103 | . . . . 5 ⊢ (𝑅 Er 𝐴 → (𝑦𝑅𝑥 ↔ 𝑥𝑅𝑦)) |
5 | vex 3418 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
6 | vex 3418 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
7 | 5, 6 | brcnv 5603 | . . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) |
8 | df-br 4930 | . . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ◡𝑅) | |
9 | 7, 8 | bitr3i 269 | . . . . 5 ⊢ (𝑦𝑅𝑥 ↔ 〈𝑥, 𝑦〉 ∈ ◡𝑅) |
10 | df-br 4930 | . . . . 5 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
11 | 4, 9, 10 | 3bitr3g 305 | . . . 4 ⊢ (𝑅 Er 𝐴 → (〈𝑥, 𝑦〉 ∈ ◡𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅)) |
12 | 11 | eqrelrdv2 5518 | . . 3 ⊢ (((Rel ◡𝑅 ∧ Rel 𝑅) ∧ 𝑅 Er 𝐴) → ◡𝑅 = 𝑅) |
13 | 2, 12 | mpanl1 687 | . 2 ⊢ ((Rel 𝑅 ∧ 𝑅 Er 𝐴) → ◡𝑅 = 𝑅) |
14 | 1, 13 | mpancom 675 | 1 ⊢ (𝑅 Er 𝐴 → ◡𝑅 = 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 〈cop 4447 class class class wbr 4929 ◡ccnv 5406 Rel wrel 5412 Er wer 8086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pr 5186 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-br 4930 df-opab 4992 df-xp 5413 df-rel 5414 df-cnv 5415 df-er 8089 |
This theorem is referenced by: errn 8111 prjspeclsp 38675 |
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