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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 2084 | . . 3 ⊢ (𝜑 → dom 𝑅 = dom 𝑅) | |
3 | iserd.2 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦𝑅𝑥) | |
4 | 3 | ex 113 | . . . . . . 7 ⊢ (𝜑 → (𝑥𝑅𝑦 → 𝑦𝑅𝑥)) |
5 | iserd.3 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)) → 𝑥𝑅𝑧) | |
6 | 5 | ex 113 | . . . . . . 7 ⊢ (𝜑 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
7 | 4, 6 | jca 300 | . . . . . 6 ⊢ (𝜑 → ((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) |
8 | 7 | alrimiv 1797 | . . . . 5 ⊢ (𝜑 → ∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) |
9 | 8 | alrimiv 1797 | . . . 4 ⊢ (𝜑 → ∀𝑦∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) |
10 | 9 | alrimiv 1797 | . . 3 ⊢ (𝜑 → ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) |
11 | dfer2 6223 | . . 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 1123 | . 2 ⊢ (𝜑 → 𝑅 Er dom 𝑅) |
13 | 12 | adantr 270 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝑅) → 𝑅 Er dom 𝑅) |
14 | simpr 108 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝑅) → 𝑥 ∈ dom 𝑅) | |
15 | 13, 14 | erref 6242 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝑅) → 𝑥𝑅𝑥) |
16 | 15 | ex 113 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ dom 𝑅 → 𝑥𝑅𝑥)) |
17 | vex 2615 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
18 | 17, 17 | breldm 4598 | . . . . . 6 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ dom 𝑅) |
19 | 16, 18 | impbid1 140 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ dom 𝑅 ↔ 𝑥𝑅𝑥)) |
20 | iserd.4 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥)) | |
21 | 19, 20 | bitr4d 189 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ dom 𝑅 ↔ 𝑥 ∈ 𝐴)) |
22 | 21 | eqrdv 2081 | . . 3 ⊢ (𝜑 → dom 𝑅 = 𝐴) |
23 | ereq2 6230 | . . 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 145 | 1 ⊢ (𝜑 → 𝑅 Er 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∀wal 1283 = wceq 1285 ∈ wcel 1434 class class class wbr 3811 dom cdm 4401 Rel wrel 4406 Er wer 6219 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2614 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-br 3812 df-opab 3866 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-er 6222 |
This theorem is referenced by: swoer 6250 eqer 6254 0er 6256 iinerm 6294 erinxp 6296 ecopover 6320 ecopoverg 6323 ener 6426 enq0er 6897 |
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