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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dfeqvrels3 | Structured version Visualization version GIF version | ||
| Description: Alternate definition of the class of equivalence relations. (Contributed by Peter Mazsa, 2-Dec-2019.) | 
| Ref | Expression | 
|---|---|
| dfeqvrels3 | ⊢ EqvRels = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥) ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dfeqvrels2 38589 | . 2 ⊢ EqvRels = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} | |
| 2 | idrefALT 6131 | . . 3 ⊢ (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥) | |
| 3 | cnvsym 6132 | . . 3 ⊢ (◡𝑟 ⊆ 𝑟 ↔ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥)) | |
| 4 | cotr 6130 | . . 3 ⊢ ((𝑟 ∘ 𝑟) ⊆ 𝑟 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) | |
| 5 | 2, 3, 4 | 3anbi123i 1156 | . 2 ⊢ ((( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ↔ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥) ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))) | 
| 6 | 1, 5 | rabbieq 3445 | 1 ⊢ EqvRels = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥) ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))} | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 ∀wal 1538 = wceq 1540 ∀wral 3061 {crab 3436 ⊆ wss 3951 class class class wbr 5143 I cid 5577 ◡ccnv 5684 dom cdm 5685 ↾ cres 5687 ∘ ccom 5689 Rels crels 38184 EqvRels ceqvrels 38198 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-rels 38486 df-ssr 38499 df-refs 38511 df-refrels 38512 df-syms 38543 df-symrels 38544 df-trs 38573 df-trrels 38574 df-eqvrels 38585 | 
| This theorem is referenced by: eleqvrels3 38594 | 
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