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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqvrelcl | Structured version Visualization version GIF version |
Description: Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 2-Jun-2019.) |
Ref | Expression |
---|---|
eqvrelcl.1 | ⊢ (𝜑 → EqvRel 𝑅) |
eqvrelcl.2 | ⊢ (𝜑 → 𝐴𝑅𝐵) |
Ref | Expression |
---|---|
eqvrelcl | ⊢ (𝜑 → 𝐴 ∈ dom 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvrelcl.1 | . . 3 ⊢ (𝜑 → EqvRel 𝑅) | |
2 | eqvrelrel 35866 | . . 3 ⊢ ( EqvRel 𝑅 → Rel 𝑅) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → Rel 𝑅) |
4 | eqvrelcl.2 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
5 | releldm 5807 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅) | |
6 | 3, 4, 5 | syl2anc 586 | 1 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 class class class wbr 5059 dom cdm 5548 Rel wrel 5553 EqvRel weqvrel 35504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-br 5060 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-refrel 35786 df-symrel 35814 df-trrel 35844 df-eqvrel 35854 |
This theorem is referenced by: eqvrelthi 35882 erim2 35945 |
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