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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 36732, eqvrelcoss3 36731 and eqvrelcoss4 36733 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.) |
Ref | Expression |
---|---|
dfcoeleqvrel | ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-coeleqvrel 36700 | . 2 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴)) | |
2 | df-coels 36538 | . . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴) | |
3 | 2 | eqvreleqi 36716 | . 2 ⊢ ( EqvRel ∼ 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴)) |
4 | 1, 3 | bitr4i 277 | 1 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 E cep 5494 ◡ccnv 5588 ↾ cres 5591 ≀ ccoss 36333 ∼ ccoels 36334 EqvRel weqvrel 36350 CoElEqvRel wcoeleqvrel 36352 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-coels 36538 df-refrel 36630 df-symrel 36658 df-trrel 36688 df-eqvrel 36698 df-coeleqvrel 36700 |
This theorem is referenced by: dfmember3 36786 |
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