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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 2037 | . 2 ⊢ (∀𝑦∀𝑥(〈𝑦, 𝑥〉 ∈ ◡𝑅 → 〈𝑦, 𝑥〉 ∈ 𝑅) ↔ ∀𝑥∀𝑦(〈𝑦, 𝑥〉 ∈ ◡𝑅 → 〈𝑦, 𝑥〉 ∈ 𝑅)) | |
2 | relcnv 5503 | . . 3 ⊢ Rel ◡𝑅 | |
3 | ssrel 5207 | . . 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 3203 | . . . . . 6 ⊢ 𝑦 ∈ V | |
6 | vex 3203 | . . . . . 6 ⊢ 𝑥 ∈ V | |
7 | 5, 6 | brcnv 5305 | . . . . 5 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦) |
8 | df-br 4654 | . . . . 5 ⊢ (𝑦◡𝑅𝑥 ↔ 〈𝑦, 𝑥〉 ∈ ◡𝑅) | |
9 | 7, 8 | bitr3i 266 | . . . 4 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑦, 𝑥〉 ∈ ◡𝑅) |
10 | df-br 4654 | . . . 4 ⊢ (𝑦𝑅𝑥 ↔ 〈𝑦, 𝑥〉 ∈ 𝑅) | |
11 | 9, 10 | imbi12i 340 | . . 3 ⊢ ((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ↔ (〈𝑦, 𝑥〉 ∈ ◡𝑅 → 〈𝑦, 𝑥〉 ∈ 𝑅)) |
12 | 11 | 2albii 1748 | . 2 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(〈𝑦, 𝑥〉 ∈ ◡𝑅 → 〈𝑦, 𝑥〉 ∈ 𝑅)) |
13 | 1, 4, 12 | 3bitr4i 292 | 1 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 ∈ wcel 1990 ⊆ wss 3574 〈cop 4183 class class class wbr 4653 ◡ccnv 5113 Rel wrel 5119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 |
This theorem is referenced by: dfer2 7743 relcnveq3 34092 relcnveq 34093 relcnveq2 34094 |
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