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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfcoeleqvrel | Structured version Visualization version GIF version |
Description: Alternate definition of the coelement equivalence relation predicate: a coelement equivalence relation is an equivalence relation on coelements. Other alternate definitions should be based on eqvrelcoss2 35888, eqvrelcoss3 35887 and eqvrelcoss4 35889 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.) |
Ref | Expression |
---|---|
dfcoeleqvrel | ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-coeleqvrel 35856 | . 2 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴)) | |
2 | df-coels 35694 | . . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴) | |
3 | 2 | eqvreleqi 35872 | . 2 ⊢ ( EqvRel ∼ 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴)) |
4 | 1, 3 | bitr4i 280 | 1 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 E cep 5457 ◡ccnv 5547 ↾ cres 5550 ≀ ccoss 35487 ∼ ccoels 35488 EqvRel weqvrel 35504 CoElEqvRel wcoeleqvrel 35506 |
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-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-coels 35694 df-refrel 35786 df-symrel 35814 df-trrel 35844 df-eqvrel 35854 df-coeleqvrel 35856 |
This theorem is referenced by: dfmember3 35942 |
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