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Mirrors > Home > MPE Home > Th. List > Mathboxes > trreleq | Structured version Visualization version GIF version |
Description: Equality theorem for the transitive relation predicate. (Contributed by Peter Mazsa, 15-Apr-2019.) (Revised by Peter Mazsa, 23-Sep-2021.) |
Ref | Expression |
---|---|
trreleq | ⊢ (𝑅 = 𝑆 → ( TrRel 𝑅 ↔ TrRel 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coideq 36733 | . . . 4 ⊢ (𝑅 = 𝑆 → (𝑅 ∘ 𝑅) = (𝑆 ∘ 𝑆)) | |
2 | id 22 | . . . 4 ⊢ (𝑅 = 𝑆 → 𝑅 = 𝑆) | |
3 | 1, 2 | sseq12d 3982 | . . 3 ⊢ (𝑅 = 𝑆 → ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ (𝑆 ∘ 𝑆) ⊆ 𝑆)) |
4 | releq 5737 | . . 3 ⊢ (𝑅 = 𝑆 → (Rel 𝑅 ↔ Rel 𝑆)) | |
5 | 3, 4 | anbi12d 632 | . 2 ⊢ (𝑅 = 𝑆 → (((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ↔ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ∧ Rel 𝑆))) |
6 | dftrrel2 37068 | . 2 ⊢ ( TrRel 𝑅 ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ Rel 𝑅)) | |
7 | dftrrel2 37068 | . 2 ⊢ ( TrRel 𝑆 ↔ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ∧ Rel 𝑆)) | |
8 | 5, 6, 7 | 3bitr4g 314 | 1 ⊢ (𝑅 = 𝑆 → ( TrRel 𝑅 ↔ TrRel 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ⊆ wss 3915 ∘ ccom 5642 Rel wrel 5643 TrRel wtrrel 36678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-trrel 37065 |
This theorem is referenced by: eqvreleq 37093 |
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