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Mirrors > Home > ILE Home > Th. List > iserd | GIF version |
Description: A reflexive, symmetric, transitive relation is an equivalence relation on its domain. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
iserd.1 | ⊢ (𝜑 → Rel 𝑅) |
iserd.2 | ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦𝑅𝑥) |
iserd.3 | ⊢ ((𝜑 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)) → 𝑥𝑅𝑧) |
iserd.4 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥)) |
Ref | Expression |
---|---|
iserd | ⊢ (𝜑 → 𝑅 Er 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iserd.1 | . . 3 ⊢ (𝜑 → Rel 𝑅) | |
2 | eqidd 2165 | . . 3 ⊢ (𝜑 → dom 𝑅 = dom 𝑅) | |
3 | iserd.2 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦𝑅𝑥) | |
4 | 3 | ex 114 | . . . . . . 7 ⊢ (𝜑 → (𝑥𝑅𝑦 → 𝑦𝑅𝑥)) |
5 | iserd.3 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)) → 𝑥𝑅𝑧) | |
6 | 5 | ex 114 | . . . . . . 7 ⊢ (𝜑 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
7 | 4, 6 | jca 304 | . . . . . 6 ⊢ (𝜑 → ((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) |
8 | 7 | alrimiv 1861 | . . . . 5 ⊢ (𝜑 → ∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) |
9 | 8 | alrimiv 1861 | . . . 4 ⊢ (𝜑 → ∀𝑦∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) |
10 | 9 | alrimiv 1861 | . . 3 ⊢ (𝜑 → ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) |
11 | dfer2 6493 | . . 3 ⊢ (𝑅 Er dom 𝑅 ↔ (Rel 𝑅 ∧ dom 𝑅 = dom 𝑅 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))) | |
12 | 1, 2, 10, 11 | syl3anbrc 1170 | . 2 ⊢ (𝜑 → 𝑅 Er dom 𝑅) |
13 | 12 | adantr 274 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝑅) → 𝑅 Er dom 𝑅) |
14 | simpr 109 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝑅) → 𝑥 ∈ dom 𝑅) | |
15 | 13, 14 | erref 6512 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝑅) → 𝑥𝑅𝑥) |
16 | 15 | ex 114 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ dom 𝑅 → 𝑥𝑅𝑥)) |
17 | vex 2724 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
18 | 17, 17 | breldm 4802 | . . . . . 6 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ dom 𝑅) |
19 | 16, 18 | impbid1 141 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ dom 𝑅 ↔ 𝑥𝑅𝑥)) |
20 | iserd.4 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥)) | |
21 | 19, 20 | bitr4d 190 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ dom 𝑅 ↔ 𝑥 ∈ 𝐴)) |
22 | 21 | eqrdv 2162 | . . 3 ⊢ (𝜑 → dom 𝑅 = 𝐴) |
23 | ereq2 6500 | . . 3 ⊢ (dom 𝑅 = 𝐴 → (𝑅 Er dom 𝑅 ↔ 𝑅 Er 𝐴)) | |
24 | 22, 23 | syl 14 | . 2 ⊢ (𝜑 → (𝑅 Er dom 𝑅 ↔ 𝑅 Er 𝐴)) |
25 | 12, 24 | mpbid 146 | 1 ⊢ (𝜑 → 𝑅 Er 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1340 = wceq 1342 ∈ wcel 2135 class class class wbr 3976 dom cdm 4598 Rel wrel 4603 Er wer 6489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-er 6492 |
This theorem is referenced by: swoer 6520 eqer 6524 0er 6526 iinerm 6564 erinxp 6566 ecopover 6590 ecopoverg 6593 ener 6736 enq0er 7367 xmeter 12977 |
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