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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.) (Proof shortened by SN, 23-Dec-2024.) Avoid ax-11 2154. (Revised by BTernaryTau, 29-Dec-2024.) |
Ref | Expression |
---|---|
cnvsym | ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 6124 | . . 3 ⊢ Rel ◡𝑅 | |
2 | ssrel3 5798 | . . 3 ⊢ (Rel ◡𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ∀𝑦∀𝑥(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑦∀𝑥(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥)) |
4 | breq1 5150 | . . . 4 ⊢ (𝑦 = 𝑧 → (𝑦◡𝑅𝑥 ↔ 𝑧◡𝑅𝑥)) | |
5 | breq1 5150 | . . . 4 ⊢ (𝑦 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑧𝑅𝑥)) | |
6 | 4, 5 | imbi12d 344 | . . 3 ⊢ (𝑦 = 𝑧 → ((𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑧◡𝑅𝑥 → 𝑧𝑅𝑥))) |
7 | breq2 5151 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑦◡𝑅𝑥 ↔ 𝑦◡𝑅𝑧)) | |
8 | breq2 5151 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑧)) | |
9 | 7, 8 | imbi12d 344 | . . 3 ⊢ (𝑥 = 𝑧 → ((𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑦◡𝑅𝑧 → 𝑦𝑅𝑧))) |
10 | 6, 9 | alcomw 2041 | . 2 ⊢ (∀𝑦∀𝑥(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥)) |
11 | vex 3481 | . . . . 5 ⊢ 𝑦 ∈ V | |
12 | vex 3481 | . . . . 5 ⊢ 𝑥 ∈ V | |
13 | 11, 12 | brcnv 5895 | . . . 4 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦) |
14 | 13 | imbi1i 349 | . . 3 ⊢ ((𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 → 𝑦𝑅𝑥)) |
15 | 14 | 2albii 1816 | . 2 ⊢ (∀𝑥∀𝑦(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) |
16 | 3, 10, 15 | 3bitri 297 | 1 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1534 ⊆ wss 3962 class class class wbr 5147 ◡ccnv 5687 Rel wrel 5693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-rel 5695 df-cnv 5696 |
This theorem is referenced by: dfer2 8744 relcnveq3 38302 relcnveq 38303 relcnveq2 38304 cnvcosseq 38418 symrelcoss2 38447 elrelscnveq3 38472 elrelscnveq 38473 elrelscnveq2 38474 dfsymrels3 38527 dfsymrel3 38531 symrefref3 38545 refsymrels3 38547 elrefsymrels3 38551 dfeqvrels3 38570 |
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