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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 38620, eqvrelcoss3 38619 and eqvrelcoss4 38621 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.) | 
| Ref | Expression | 
|---|---|
| dfcoeleqvrel | ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-coeleqvrel 38588 | . 2 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴)) | |
| 2 | df-coels 38413 | . . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴) | |
| 3 | 2 | eqvreleqi 38604 | . 2 ⊢ ( EqvRel ∼ 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴)) | 
| 4 | 1, 3 | bitr4i 278 | 1 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 E cep 5583 ◡ccnv 5684 ↾ cres 5687 ≀ ccoss 38182 ∼ ccoels 38183 EqvRel weqvrel 38199 CoElEqvRel wcoeleqvrel 38201 | 
| 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-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-coels 38413 df-refrel 38513 df-symrel 38545 df-trrel 38575 df-eqvrel 38586 df-coeleqvrel 38588 | 
| This theorem is referenced by: dfcomember3 38675 | 
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