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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 2729 | . . 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 1923 | . . . . 5 ⊢ (𝜑 → ∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) |
9 | 8 | alrimiv 1923 | . . . 4 ⊢ (𝜑 → ∀𝑦∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) |
10 | 9 | alrimiv 1923 | . . 3 ⊢ (𝜑 → ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) |
11 | dfer2 8726 | . . 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 1341 | . 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 8745 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝑅) → 𝑥𝑅𝑥) |
16 | 15 | ex 412 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ dom 𝑅 → 𝑥𝑅𝑥)) |
17 | vex 3475 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
18 | 17, 17 | breldm 5911 | . . . . . 6 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ dom 𝑅) |
19 | 16, 18 | impbid1 224 | . . . . 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 2726 | . . 3 ⊢ (𝜑 → dom 𝑅 = 𝐴) |
23 | ereq2 8733 | . . 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 231 | 1 ⊢ (𝜑 → 𝑅 Er 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1532 = wceq 1534 ∈ wcel 2099 class class class wbr 5148 dom cdm 5678 Rel wrel 5683 Er wer 8722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-er 8725 |
This theorem is referenced by: iseri 8752 iseriALT 8753 swoer 8755 iiner 8808 erinxp 8810 cicer 17789 eqger 19133 gaorber 19259 efgrelexlemb 19705 efgcpbllemb 19710 xmeter 24352 ercgrg 28334 erler 32992 metider 33495 prjsper 42032 |
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