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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eleqvrels2 | Structured version Visualization version GIF version |
Description: Element of the class of equivalence relations. (Contributed by Peter Mazsa, 24-Aug-2021.) |
Ref | Expression |
---|---|
eleqvrels2 | ⊢ (𝑅 ∈ EqvRels ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) ∧ 𝑅 ∈ Rels )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfeqvrels2 38116 | . 2 ⊢ EqvRels = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} | |
2 | dmeq 5900 | . . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅) | |
3 | 2 | reseq2d 5979 | . . . 4 ⊢ (𝑟 = 𝑅 → ( I ↾ dom 𝑟) = ( I ↾ dom 𝑅)) |
4 | id 22 | . . . 4 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
5 | 3, 4 | sseq12d 4006 | . . 3 ⊢ (𝑟 = 𝑅 → (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅)) |
6 | cnveq 5870 | . . . 4 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅) | |
7 | 6, 4 | sseq12d 4006 | . . 3 ⊢ (𝑟 = 𝑅 → (◡𝑟 ⊆ 𝑟 ↔ ◡𝑅 ⊆ 𝑅)) |
8 | coideq 37774 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 ∘ 𝑟) = (𝑅 ∘ 𝑅)) | |
9 | 8, 4 | sseq12d 4006 | . . 3 ⊢ (𝑟 = 𝑅 → ((𝑟 ∘ 𝑟) ⊆ 𝑟 ↔ (𝑅 ∘ 𝑅) ⊆ 𝑅)) |
10 | 5, 7, 9 | 3anbi123d 1432 | . 2 ⊢ (𝑟 = 𝑅 → ((( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ↔ (( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅))) |
11 | 1, 10 | rabeqel 37782 | 1 ⊢ (𝑅 ∈ EqvRels ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) ∧ 𝑅 ∈ Rels )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⊆ wss 3939 I cid 5569 ◡ccnv 5671 dom cdm 5672 ↾ cres 5674 ∘ ccom 5676 Rels crels 37707 EqvRels ceqvrels 37721 |
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 5294 ax-nul 5301 ax-pr 5423 |
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 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-br 5144 df-opab 5206 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-rels 38013 df-ssr 38026 df-refs 38038 df-refrels 38039 df-syms 38070 df-symrels 38071 df-trs 38100 df-trrels 38101 df-eqvrels 38112 |
This theorem is referenced by: eleqvrelsrel 38122 |
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