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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqvreltr | Structured version Visualization version GIF version |
Description: An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 2-Jun-2019.) |
Ref | Expression |
---|---|
eqvreltr.1 | ⊢ (𝜑 → EqvRel 𝑅) |
Ref | Expression |
---|---|
eqvreltr | ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvreltr.1 | . . . . . . 7 ⊢ (𝜑 → EqvRel 𝑅) | |
2 | eqvrelrel 34887 | . . . . . . 7 ⊢ ( EqvRel 𝑅 → Rel 𝑅) | |
3 | 1, 2 | syl 17 | . . . . . 6 ⊢ (𝜑 → Rel 𝑅) |
4 | simpr 479 | . . . . . 6 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐵𝑅𝐶) | |
5 | brrelex1 5390 | . . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐶) → 𝐵 ∈ V) | |
6 | 3, 4, 5 | syl2an 591 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐵 ∈ V) |
7 | simpr 479 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) | |
8 | breq2 4877 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵)) | |
9 | breq1 4876 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝐶 ↔ 𝐵𝑅𝐶)) | |
10 | 8, 9 | anbi12d 626 | . . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐴𝑅𝑥 ∧ 𝑥𝑅𝐶) ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶))) |
11 | 6, 7, 10 | elabd 3573 | . . . 4 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥𝑅𝐶)) |
12 | simpl 476 | . . . . . 6 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐵) | |
13 | brrelex1 5390 | . . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V) | |
14 | 3, 12, 13 | syl2an 591 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐴 ∈ V) |
15 | brrelex2 5391 | . . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐶) → 𝐶 ∈ V) | |
16 | 3, 4, 15 | syl2an 591 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐶 ∈ V) |
17 | brcog 5521 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴(𝑅 ∘ 𝑅)𝐶 ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥𝑅𝐶))) | |
18 | 14, 16, 17 | syl2anc 581 | . . . 4 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → (𝐴(𝑅 ∘ 𝑅)𝐶 ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥𝑅𝐶))) |
19 | 11, 18 | mpbird 249 | . . 3 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐴(𝑅 ∘ 𝑅)𝐶) |
20 | 19 | ex 403 | . 2 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴(𝑅 ∘ 𝑅)𝐶)) |
21 | dfeqvrel2 34882 | . . . . . 6 ⊢ ( EqvRel 𝑅 ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) ∧ Rel 𝑅)) | |
22 | 21 | simplbi 493 | . . . . 5 ⊢ ( EqvRel 𝑅 → (( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅)) |
23 | 22 | simp3d 1180 | . . . 4 ⊢ ( EqvRel 𝑅 → (𝑅 ∘ 𝑅) ⊆ 𝑅) |
24 | 1, 23 | syl 17 | . . 3 ⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅) |
25 | 24 | ssbrd 4916 | . 2 ⊢ (𝜑 → (𝐴(𝑅 ∘ 𝑅)𝐶 → 𝐴𝑅𝐶)) |
26 | 20, 25 | syld 47 | 1 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∃wex 1880 ∈ wcel 2166 Vcvv 3414 ⊆ wss 3798 class class class wbr 4873 I cid 5249 ◡ccnv 5341 dom cdm 5342 ↾ cres 5344 ∘ ccom 5346 Rel wrel 5347 EqvRel weqvrel 34541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4874 df-opab 4936 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-refrel 34810 df-symrel 34838 df-trrel 34868 df-eqvrel 34878 |
This theorem is referenced by: eqvreltrd 34898 eqvrelth 34901 |
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