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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 38015, eqvrelcoss3 38014 and eqvrelcoss4 38016 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.) |
Ref | Expression |
---|---|
dfcoeleqvrel | ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-coeleqvrel 37983 | . 2 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴)) | |
2 | df-coels 37808 | . . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴) | |
3 | 2 | eqvreleqi 37999 | . 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 205 E cep 5575 ◡ccnv 5671 ↾ cres 5674 ≀ ccoss 37570 ∼ ccoels 37571 EqvRel weqvrel 37587 CoElEqvRel wcoeleqvrel 37589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-coels 37808 df-refrel 37908 df-symrel 37940 df-trrel 37970 df-eqvrel 37981 df-coeleqvrel 37983 |
This theorem is referenced by: dfcomember3 38070 |
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