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Mirrors > Home > MPE Home > Th. List > cnvsym | Structured version Visualization version GIF version |
Description: Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
cnvsym | ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alcom 2024 | . 2 ⊢ (∀𝑦∀𝑥(〈𝑦, 𝑥〉 ∈ ◡𝑅 → 〈𝑦, 𝑥〉 ∈ 𝑅) ↔ ∀𝑥∀𝑦(〈𝑦, 𝑥〉 ∈ ◡𝑅 → 〈𝑦, 𝑥〉 ∈ 𝑅)) | |
2 | relcnv 5422 | . . 3 ⊢ Rel ◡𝑅 | |
3 | ssrel 5130 | . . 3 ⊢ (Rel ◡𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ∀𝑦∀𝑥(〈𝑦, 𝑥〉 ∈ ◡𝑅 → 〈𝑦, 𝑥〉 ∈ 𝑅))) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑦∀𝑥(〈𝑦, 𝑥〉 ∈ ◡𝑅 → 〈𝑦, 𝑥〉 ∈ 𝑅)) |
5 | vex 3176 | . . . . . 6 ⊢ 𝑦 ∈ V | |
6 | vex 3176 | . . . . . 6 ⊢ 𝑥 ∈ V | |
7 | 5, 6 | brcnv 5227 | . . . . 5 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦) |
8 | df-br 4584 | . . . . 5 ⊢ (𝑦◡𝑅𝑥 ↔ 〈𝑦, 𝑥〉 ∈ ◡𝑅) | |
9 | 7, 8 | bitr3i 265 | . . . 4 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑦, 𝑥〉 ∈ ◡𝑅) |
10 | df-br 4584 | . . . 4 ⊢ (𝑦𝑅𝑥 ↔ 〈𝑦, 𝑥〉 ∈ 𝑅) | |
11 | 9, 10 | imbi12i 339 | . . 3 ⊢ ((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ↔ (〈𝑦, 𝑥〉 ∈ ◡𝑅 → 〈𝑦, 𝑥〉 ∈ 𝑅)) |
12 | 11 | 2albii 1738 | . 2 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(〈𝑦, 𝑥〉 ∈ ◡𝑅 → 〈𝑦, 𝑥〉 ∈ 𝑅)) |
13 | 1, 4, 12 | 3bitr4i 291 | 1 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 ∈ wcel 1977 ⊆ wss 3540 〈cop 4131 class class class wbr 4583 ◡ccnv 5037 Rel wrel 5043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 |
This theorem is referenced by: dfer2 7630 |
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