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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 5743 | . . . 4 ⊢ (𝑅 = 𝑆 → ◡𝑅 = ◡𝑆) | |
2 | id 22 | . . . 4 ⊢ (𝑅 = 𝑆 → 𝑅 = 𝑆) | |
3 | 1, 2 | sseq12d 3999 | . . 3 ⊢ (𝑅 = 𝑆 → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑆 ⊆ 𝑆)) |
4 | releq 5650 | . . 3 ⊢ (𝑅 = 𝑆 → (Rel 𝑅 ↔ Rel 𝑆)) | |
5 | 3, 4 | anbi12d 632 | . 2 ⊢ (𝑅 = 𝑆 → ((◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅) ↔ (◡𝑆 ⊆ 𝑆 ∧ Rel 𝑆))) |
6 | dfsymrel2 35784 | . 2 ⊢ ( SymRel 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅)) | |
7 | dfsymrel2 35784 | . 2 ⊢ ( SymRel 𝑆 ↔ (◡𝑆 ⊆ 𝑆 ∧ Rel 𝑆)) | |
8 | 5, 6, 7 | 3bitr4g 316 | 1 ⊢ (𝑅 = 𝑆 → ( SymRel 𝑅 ↔ SymRel 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ⊆ wss 3935 ◡ccnv 5553 Rel wrel 5559 SymRel wsymrel 35464 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-opab 5128 df-xp 5560 df-rel 5561 df-cnv 5562 df-dm 5564 df-rn 5565 df-res 5566 df-symrel 35779 |
This theorem is referenced by: eqvreleq 35836 |
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