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Mirrors > Home > MPE Home > Th. List > Mathboxes > symreleq | Structured version Visualization version GIF version |
Description: Equality theorem for symmetric relation. (Contributed by Peter Mazsa, 15-Apr-2019.) (Revised by Peter Mazsa, 23-Sep-2021.) |
Ref | Expression |
---|---|
symreleq | ⊢ (𝑅 = 𝑆 → ( SymRel 𝑅 ↔ SymRel 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnveq 5630 | . . . 4 ⊢ (𝑅 = 𝑆 → ◡𝑅 = ◡𝑆) | |
2 | id 22 | . . . 4 ⊢ (𝑅 = 𝑆 → 𝑅 = 𝑆) | |
3 | 1, 2 | sseq12d 3921 | . . 3 ⊢ (𝑅 = 𝑆 → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑆 ⊆ 𝑆)) |
4 | releq 5537 | . . 3 ⊢ (𝑅 = 𝑆 → (Rel 𝑅 ↔ Rel 𝑆)) | |
5 | 3, 4 | anbi12d 630 | . 2 ⊢ (𝑅 = 𝑆 → ((◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅) ↔ (◡𝑆 ⊆ 𝑆 ∧ Rel 𝑆))) |
6 | dfsymrel2 35316 | . 2 ⊢ ( SymRel 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅)) | |
7 | dfsymrel2 35316 | . 2 ⊢ ( SymRel 𝑆 ↔ (◡𝑆 ⊆ 𝑆 ∧ Rel 𝑆)) | |
8 | 5, 6, 7 | 3bitr4g 315 | 1 ⊢ (𝑅 = 𝑆 → ( SymRel 𝑅 ↔ SymRel 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ⊆ wss 3859 ◡ccnv 5442 Rel wrel 5448 SymRel wsymrel 34997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-br 4963 df-opab 5025 df-xp 5449 df-rel 5450 df-cnv 5451 df-dm 5453 df-rn 5454 df-res 5455 df-symrel 35311 |
This theorem is referenced by: eqvreleq 35368 |
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