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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 37980 | . . . 4 ⊢ ( EqvRel 𝑅 → Rel 𝑅) | |
4 | releldmb 5939 | . . . 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 231 | . 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 37992 | . 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝐴) |
10 | 6, 9 | exlimddv 1930 | 1 ⊢ (𝜑 → 𝐴𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∃wex 1773 ∈ wcel 2098 class class class wbr 5141 dom cdm 5669 Rel wrel 5674 EqvRel weqvrel 37573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-refrel 37895 df-symrel 37927 df-trrel 37957 df-eqvrel 37968 |
This theorem is referenced by: eqvrelth 37994 |
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