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| Mirrors > Home > MPE Home > Th. List > Mathboxes > brers | Structured version Visualization version GIF version | ||
| Description: Binary equivalence relation with natural domain, see the comment of df-ers 38664. (Contributed by Peter Mazsa, 23-Jul-2021.) |
| Ref | Expression |
|---|---|
| brers | ⊢ (𝐴 ∈ 𝑉 → (𝑅 Ers 𝐴 ↔ (𝑅 ∈ EqvRels ∧ 𝑅 DomainQss 𝐴))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ers 38664 | . 2 ⊢ Ers = ( DomainQss ↾ EqvRels ) | |
| 2 | 1 | eqres 38341 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝑅 Ers 𝐴 ↔ (𝑅 ∈ EqvRels ∧ 𝑅 DomainQss 𝐴))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 class class class wbr 5143 EqvRels ceqvrels 38198 DomainQss cdmqss 38205 Ers cers 38207 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-res 5697 df-ers 38664 |
| This theorem is referenced by: brerser 38678 |
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