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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 2188 | . . 3 ⊢ (𝜑 → dom 𝑅 = dom 𝑅) | |
3 | iserd.2 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦𝑅𝑥) | |
4 | 3 | ex 115 | . . . . . . 7 ⊢ (𝜑 → (𝑥𝑅𝑦 → 𝑦𝑅𝑥)) |
5 | iserd.3 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)) → 𝑥𝑅𝑧) | |
6 | 5 | ex 115 | . . . . . . 7 ⊢ (𝜑 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
7 | 4, 6 | jca 306 | . . . . . 6 ⊢ (𝜑 → ((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) |
8 | 7 | alrimiv 1884 | . . . . 5 ⊢ (𝜑 → ∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) |
9 | 8 | alrimiv 1884 | . . . 4 ⊢ (𝜑 → ∀𝑦∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) |
10 | 9 | alrimiv 1884 | . . 3 ⊢ (𝜑 → ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) |
11 | dfer2 6549 | . . 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 1182 | . 2 ⊢ (𝜑 → 𝑅 Er dom 𝑅) |
13 | 12 | adantr 276 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝑅) → 𝑅 Er dom 𝑅) |
14 | simpr 110 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝑅) → 𝑥 ∈ dom 𝑅) | |
15 | 13, 14 | erref 6568 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝑅) → 𝑥𝑅𝑥) |
16 | 15 | ex 115 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ dom 𝑅 → 𝑥𝑅𝑥)) |
17 | vex 2752 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
18 | 17, 17 | breldm 4843 | . . . . . 6 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ dom 𝑅) |
19 | 16, 18 | impbid1 142 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ dom 𝑅 ↔ 𝑥𝑅𝑥)) |
20 | iserd.4 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥)) | |
21 | 19, 20 | bitr4d 191 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ dom 𝑅 ↔ 𝑥 ∈ 𝐴)) |
22 | 21 | eqrdv 2185 | . . 3 ⊢ (𝜑 → dom 𝑅 = 𝐴) |
23 | ereq2 6556 | . . 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 147 | 1 ⊢ (𝜑 → 𝑅 Er 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∀wal 1361 = wceq 1363 ∈ wcel 2158 class class class wbr 4015 dom cdm 4638 Rel wrel 4643 Er wer 6545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-br 4016 df-opab 4077 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-er 6548 |
This theorem is referenced by: swoer 6576 eqer 6580 0er 6582 iinerm 6620 erinxp 6622 ecopover 6646 ecopoverg 6649 ener 6792 enq0er 7447 eqger 13115 xmeter 14207 |
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