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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 37088 | . . . 4 ⊢ ( EqvRel 𝑅 → Rel 𝑅) | |
4 | releldmb 5906 | . . . 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 482 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → EqvRel 𝑅) |
8 | simpr 486 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝑥) | |
9 | 7, 8, 8 | eqvreltr4d 37100 | . 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝐴) |
10 | 6, 9 | exlimddv 1939 | 1 ⊢ (𝜑 → 𝐴𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∃wex 1782 ∈ wcel 2107 class class class wbr 5110 dom cdm 5638 Rel wrel 5643 EqvRel weqvrel 36680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-refrel 37003 df-symrel 37035 df-trrel 37065 df-eqvrel 37076 |
This theorem is referenced by: eqvrelth 37102 |
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