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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqvrelref | Structured version Visualization version GIF version |
Description: An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 2-Jun-2019.) |
Ref | Expression |
---|---|
eqvrelref.1 | ⊢ (𝜑 → EqvRel 𝑅) |
eqvrelref.2 | ⊢ (𝜑 → 𝐴 ∈ dom 𝑅) |
Ref | Expression |
---|---|
eqvrelref | ⊢ (𝜑 → 𝐴𝑅𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvrelref.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅) | |
2 | eqvrelref.1 | . . . 4 ⊢ (𝜑 → EqvRel 𝑅) | |
3 | eqvrelrel 38579 | . . . 4 ⊢ ( EqvRel 𝑅 → Rel 𝑅) | |
4 | releldmb 5960 | . . . 4 ⊢ (Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) | |
5 | 2, 3, 4 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) |
6 | 1, 5 | mpbid 232 | . 2 ⊢ (𝜑 → ∃𝑥 𝐴𝑅𝑥) |
7 | 2 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → EqvRel 𝑅) |
8 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝑥) | |
9 | 7, 8, 8 | eqvreltr4d 38591 | . 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝐴) |
10 | 6, 9 | exlimddv 1933 | 1 ⊢ (𝜑 → 𝐴𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∃wex 1776 ∈ wcel 2106 class class class wbr 5148 dom cdm 5689 Rel wrel 5694 EqvRel weqvrel 38179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-refrel 38494 df-symrel 38526 df-trrel 38556 df-eqvrel 38567 |
This theorem is referenced by: eqvrelth 38593 |
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