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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 37110, eqvrelcoss3 37109 and eqvrelcoss4 37111 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.) |
Ref | Expression |
---|---|
dfcoeleqvrel | ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-coeleqvrel 37078 | . 2 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴)) | |
2 | df-coels 36903 | . . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴) | |
3 | 2 | eqvreleqi 37094 | . 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 5541 ◡ccnv 5637 ↾ cres 5640 ≀ ccoss 36663 ∼ ccoels 36664 EqvRel weqvrel 36680 CoElEqvRel wcoeleqvrel 36682 |
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-10 2138 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
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-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-coels 36903 df-refrel 37003 df-symrel 37035 df-trrel 37065 df-eqvrel 37076 df-coeleqvrel 37078 |
This theorem is referenced by: dfcomember3 37165 |
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