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| Mirrors > Home > MPE Home > Th. List > iserd | Structured version Visualization version 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 2737 | . . 3 ⊢ (𝜑 → dom 𝑅 = dom 𝑅) | |
| 3 | iserd.2 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦𝑅𝑥) | |
| 4 | 3 | ex 412 | . . . . . . 7 ⊢ (𝜑 → (𝑥𝑅𝑦 → 𝑦𝑅𝑥)) | 
| 5 | iserd.3 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)) → 𝑥𝑅𝑧) | |
| 6 | 5 | ex 412 | . . . . . . 7 ⊢ (𝜑 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) | 
| 7 | 4, 6 | jca 511 | . . . . . 6 ⊢ (𝜑 → ((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) | 
| 8 | 7 | alrimiv 1926 | . . . . 5 ⊢ (𝜑 → ∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) | 
| 9 | 8 | alrimiv 1926 | . . . 4 ⊢ (𝜑 → ∀𝑦∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) | 
| 10 | 9 | alrimiv 1926 | . . 3 ⊢ (𝜑 → ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) | 
| 11 | dfer2 8747 | . . 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 1343 | . 2 ⊢ (𝜑 → 𝑅 Er dom 𝑅) | 
| 13 | 12 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝑅) → 𝑅 Er dom 𝑅) | 
| 14 | simpr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝑅) → 𝑥 ∈ dom 𝑅) | |
| 15 | 13, 14 | erref 8766 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝑅) → 𝑥𝑅𝑥) | 
| 16 | 15 | ex 412 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ dom 𝑅 → 𝑥𝑅𝑥)) | 
| 17 | vex 3483 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
| 18 | 17, 17 | breldm 5918 | . . . . . 6 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ dom 𝑅) | 
| 19 | 16, 18 | impbid1 225 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ dom 𝑅 ↔ 𝑥𝑅𝑥)) | 
| 20 | iserd.4 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥)) | |
| 21 | 19, 20 | bitr4d 282 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ dom 𝑅 ↔ 𝑥 ∈ 𝐴)) | 
| 22 | 21 | eqrdv 2734 | . . 3 ⊢ (𝜑 → dom 𝑅 = 𝐴) | 
| 23 | ereq2 8754 | . . 3 ⊢ (dom 𝑅 = 𝐴 → (𝑅 Er dom 𝑅 ↔ 𝑅 Er 𝐴)) | |
| 24 | 22, 23 | syl 17 | . 2 ⊢ (𝜑 → (𝑅 Er dom 𝑅 ↔ 𝑅 Er 𝐴)) | 
| 25 | 12, 24 | mpbid 232 | 1 ⊢ (𝜑 → 𝑅 Er 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1537 = wceq 1539 ∈ wcel 2107 class class class wbr 5142 dom cdm 5684 Rel wrel 5689 Er wer 8743 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-er 8746 | 
| This theorem is referenced by: iseri 8773 iseriALT 8774 swoer 8777 iiner 8830 erinxp 8832 cicer 17851 eqger 19197 gaorber 19327 efgrelexlemb 19769 efgcpbllemb 19774 xmeter 24444 ercgrg 28526 erler 33270 metider 33894 prjsper 42623 | 
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