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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqvrelthi | Structured version Visualization version GIF version |
Description: Basic property of equivalence relations. Part of Lemma 3N of [Enderton] p. 57. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.) (Revised by Peter Mazsa, 2-Jun-2019.) |
Ref | Expression |
---|---|
eqvrelthi.1 | ⊢ (𝜑 → EqvRel 𝑅) |
eqvrelthi.2 | ⊢ (𝜑 → 𝐴𝑅𝐵) |
Ref | Expression |
---|---|
eqvrelthi | ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvrelthi.2 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
2 | eqvrelthi.1 | . . 3 ⊢ (𝜑 → EqvRel 𝑅) | |
3 | 2, 1 | eqvrelcl 38310 | . . 3 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅) |
4 | 2, 3 | eqvrelth 38309 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅)) |
5 | 1, 4 | mpbid 231 | 1 ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 class class class wbr 5153 [cec 8732 EqvRel weqvrel 37893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-br 5154 df-opab 5216 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ec 8736 df-refrel 38210 df-symrel 38242 df-trrel 38272 df-eqvrel 38283 |
This theorem is referenced by: eqvreldisj 38312 eqvrelqsel 38314 |
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