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Mirrors > Home > MPE Home > Th. List > Mathboxes > elrelscnveq2 | Structured version Visualization version GIF version |
Description: Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.) |
Ref | Expression |
---|---|
elrelscnveq2 | ⊢ (𝑅 ∈ Rels → (◡𝑅 = 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvsym 5974 | . . . 4 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝑅 ∈ Rels → (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))) |
3 | cnvsym 5974 | . . . . 5 ⊢ (◡◡𝑅 ⊆ ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥◡𝑅𝑦 → 𝑦◡𝑅𝑥)) | |
4 | elrelsrelim 35743 | . . . . . . 7 ⊢ (𝑅 ∈ Rels → Rel 𝑅) | |
5 | dfrel2 6046 | . . . . . . 7 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
6 | 4, 5 | sylib 220 | . . . . . 6 ⊢ (𝑅 ∈ Rels → ◡◡𝑅 = 𝑅) |
7 | 6 | sseq1d 3998 | . . . . 5 ⊢ (𝑅 ∈ Rels → (◡◡𝑅 ⊆ ◡𝑅 ↔ 𝑅 ⊆ ◡𝑅)) |
8 | 3, 7 | syl5rbbr 288 | . . . 4 ⊢ (𝑅 ∈ Rels → (𝑅 ⊆ ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥◡𝑅𝑦 → 𝑦◡𝑅𝑥))) |
9 | relbrcnvg 5968 | . . . . . . 7 ⊢ (Rel 𝑅 → (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)) | |
10 | 4, 9 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ Rels → (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)) |
11 | relbrcnvg 5968 | . . . . . . 7 ⊢ (Rel 𝑅 → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)) | |
12 | 4, 11 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ Rels → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)) |
13 | 10, 12 | imbi12d 347 | . . . . 5 ⊢ (𝑅 ∈ Rels → ((𝑥◡𝑅𝑦 → 𝑦◡𝑅𝑥) ↔ (𝑦𝑅𝑥 → 𝑥𝑅𝑦))) |
14 | 13 | 2albidv 1924 | . . . 4 ⊢ (𝑅 ∈ Rels → (∀𝑥∀𝑦(𝑥◡𝑅𝑦 → 𝑦◡𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑦𝑅𝑥 → 𝑥𝑅𝑦))) |
15 | 8, 14 | bitrd 281 | . . 3 ⊢ (𝑅 ∈ Rels → (𝑅 ⊆ ◡𝑅 ↔ ∀𝑥∀𝑦(𝑦𝑅𝑥 → 𝑥𝑅𝑦))) |
16 | 2, 15 | anbi12d 632 | . 2 ⊢ (𝑅 ∈ Rels → ((◡𝑅 ⊆ 𝑅 ∧ 𝑅 ⊆ ◡𝑅) ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥∀𝑦(𝑦𝑅𝑥 → 𝑥𝑅𝑦)))) |
17 | eqss 3982 | . 2 ⊢ (◡𝑅 = 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ 𝑅 ⊆ ◡𝑅)) | |
18 | 2albiim 1891 | . 2 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥∀𝑦(𝑦𝑅𝑥 → 𝑥𝑅𝑦))) | |
19 | 16, 17, 18 | 3bitr4g 316 | 1 ⊢ (𝑅 ∈ Rels → (◡𝑅 = 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 class class class wbr 5066 ◡ccnv 5554 Rel wrel 5560 Rels crels 35470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-rels 35740 |
This theorem is referenced by: elrelscnveq4 35749 dfsymrels5 35799 |
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