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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 38644. (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 38644 | . 2 ⊢ Ers = ( DomainQss ↾ EqvRels ) | |
2 | 1 | eqres 38321 | 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 2105 class class class wbr 5147 EqvRels ceqvrels 38177 DomainQss cdmqss 38184 Ers cers 38186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-res 5700 df-ers 38644 |
This theorem is referenced by: brerser 38658 |
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