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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 38594 | . . 3 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅) |
4 | 2, 3 | eqvrelth 38593 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅)) |
5 | 1, 4 | mpbid 232 | 1 ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 class class class wbr 5148 [cec 8742 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-10 2139 ax-11 2155 ax-12 2175 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-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 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-ima 5702 df-ec 8746 df-refrel 38494 df-symrel 38526 df-trrel 38556 df-eqvrel 38567 |
This theorem is referenced by: eqvreldisj 38596 eqvrelqsel 38598 |
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