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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 37471. (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 37471 | . 2 ⊢ Ers = ( DomainQss ↾ EqvRels ) | |
2 | 1 | eqres 37147 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝑅 Ers 𝐴 ↔ (𝑅 ∈ EqvRels ∧ 𝑅 DomainQss 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 class class class wbr 5147 EqvRels ceqvrels 36997 DomainQss cdmqss 37004 Ers cers 37006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-xp 5681 df-res 5687 df-ers 37471 |
This theorem is referenced by: brerser 37485 |
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