| Mathbox for Peter Mazsa |
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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 38598 | . . . . . . 7 ⊢ ( EqvRel 𝑅 → Rel 𝑅) | |
| 3 | 1, 2 | syl 17 | . . . . . 6 ⊢ (𝜑 → Rel 𝑅) |
| 4 | simpr 484 | . . . . . 6 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐵𝑅𝐶) | |
| 5 | brrelex1 5738 | . . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐶) → 𝐵 ∈ V) | |
| 6 | 3, 4, 5 | syl2an 596 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐵 ∈ V) |
| 7 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) | |
| 8 | breq2 5147 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵)) | |
| 9 | breq1 5146 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝐶 ↔ 𝐵𝑅𝐶)) | |
| 10 | 8, 9 | anbi12d 632 | . . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐴𝑅𝑥 ∧ 𝑥𝑅𝐶) ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶))) |
| 11 | 6, 7, 10 | spcedv 3598 | . . . 4 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥𝑅𝐶)) |
| 12 | simpl 482 | . . . . . 6 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐵) | |
| 13 | brrelex1 5738 | . . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V) | |
| 14 | 3, 12, 13 | syl2an 596 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐴 ∈ V) |
| 15 | brrelex2 5739 | . . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐶) → 𝐶 ∈ V) | |
| 16 | 3, 4, 15 | syl2an 596 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐶 ∈ V) |
| 17 | brcog 5877 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴(𝑅 ∘ 𝑅)𝐶 ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥𝑅𝐶))) | |
| 18 | 14, 16, 17 | syl2anc 584 | . . . 4 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → (𝐴(𝑅 ∘ 𝑅)𝐶 ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥𝑅𝐶))) |
| 19 | 11, 18 | mpbird 257 | . . 3 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐴(𝑅 ∘ 𝑅)𝐶) |
| 20 | 19 | ex 412 | . 2 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴(𝑅 ∘ 𝑅)𝐶)) |
| 21 | dfeqvrel2 38591 | . . . . . 6 ⊢ ( EqvRel 𝑅 ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) ∧ Rel 𝑅)) | |
| 22 | 21 | simplbi 497 | . . . . 5 ⊢ ( EqvRel 𝑅 → (( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅)) |
| 23 | 22 | simp3d 1145 | . . . 4 ⊢ ( EqvRel 𝑅 → (𝑅 ∘ 𝑅) ⊆ 𝑅) |
| 24 | 1, 23 | syl 17 | . . 3 ⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅) |
| 25 | 24 | ssbrd 5186 | . 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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 I cid 5577 ◡ccnv 5684 dom cdm 5685 ↾ cres 5687 ∘ ccom 5689 Rel wrel 5690 EqvRel weqvrel 38199 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-refrel 38513 df-symrel 38545 df-trrel 38575 df-eqvrel 38586 |
| This theorem is referenced by: eqvreltrd 38609 eqvrelth 38612 |
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