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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 5876 | . . . 4 ⊢ (𝑅 = 𝑆 → ◡𝑅 = ◡𝑆) | |
2 | id 22 | . . . 4 ⊢ (𝑅 = 𝑆 → 𝑅 = 𝑆) | |
3 | 1, 2 | sseq12d 4010 | . . 3 ⊢ (𝑅 = 𝑆 → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑆 ⊆ 𝑆)) |
4 | releq 5778 | . . 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 38153 | . 2 ⊢ ( SymRel 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅)) | |
7 | dfsymrel2 38153 | . 2 ⊢ ( SymRel 𝑆 ↔ (◡𝑆 ⊆ 𝑆 ∧ Rel 𝑆)) | |
8 | 5, 6, 7 | 3bitr4g 313 | 1 ⊢ (𝑅 = 𝑆 → ( SymRel 𝑅 ↔ SymRel 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ⊆ wss 3944 ◡ccnv 5677 Rel wrel 5683 SymRel wsymrel 37793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-xp 5684 df-rel 5685 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-symrel 38148 |
This theorem is referenced by: eqvreleq 38206 |
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