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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 35930. (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 35930 | . 2 ⊢ Ers = ( DomainQss ↾ EqvRels ) | |
2 | 1 | eqres 35630 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝑅 Ers 𝐴 ↔ (𝑅 ∈ EqvRels ∧ 𝑅 DomainQss 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2113 class class class wbr 5059 EqvRels ceqvrels 35502 DomainQss cdmqss 35509 Ers cers 35511 |
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-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-xp 5554 df-res 5560 df-ers 35930 |
This theorem is referenced by: brerser 35943 |
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