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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 2157. (Revised by BTernaryTau, 29-Dec-2024.) |
| Ref | Expression |
|---|---|
| cnvsym | ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | relcnv 6122 | . . 3 ⊢ Rel ◡𝑅 | |
| 2 | ssrel3 5796 | . . 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 5146 | . . . 4 ⊢ (𝑦 = 𝑧 → (𝑦◡𝑅𝑥 ↔ 𝑧◡𝑅𝑥)) | |
| 5 | breq1 5146 | . . . 4 ⊢ (𝑦 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑧𝑅𝑥)) | |
| 6 | 4, 5 | imbi12d 344 | . . 3 ⊢ (𝑦 = 𝑧 → ((𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑧◡𝑅𝑥 → 𝑧𝑅𝑥))) |
| 7 | breq2 5147 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑦◡𝑅𝑥 ↔ 𝑦◡𝑅𝑧)) | |
| 8 | breq2 5147 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑧)) | |
| 9 | 7, 8 | imbi12d 344 | . . 3 ⊢ (𝑥 = 𝑧 → ((𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑦◡𝑅𝑧 → 𝑦𝑅𝑧))) |
| 10 | 6, 9 | alcomw 2044 | . 2 ⊢ (∀𝑦∀𝑥(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥)) |
| 11 | vex 3484 | . . . . 5 ⊢ 𝑦 ∈ V | |
| 12 | vex 3484 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 13 | 11, 12 | brcnv 5893 | . . . 4 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦) |
| 14 | 13 | imbi1i 349 | . . 3 ⊢ ((𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 → 𝑦𝑅𝑥)) |
| 15 | 14 | 2albii 1820 | . 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 1538 ⊆ wss 3951 class class class wbr 5143 ◡ccnv 5684 Rel wrel 5690 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 |
| This theorem is referenced by: dfer2 8746 relcnveq3 38322 relcnveq 38323 relcnveq2 38324 cnvcosseq 38438 symrelcoss2 38467 elrelscnveq3 38492 elrelscnveq 38493 elrelscnveq2 38494 dfsymrels3 38547 dfsymrel3 38551 symrefref3 38565 refsymrels3 38567 elrefsymrels3 38571 dfeqvrels3 38590 |
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