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Mirrors > Home > MPE Home > Th. List > Mathboxes > relcoels | Structured version Visualization version GIF version |
Description: Coelements on 𝐴 is a relation. (Contributed by Peter Mazsa, 5-Oct-2021.) |
Ref | Expression |
---|---|
relcoels | ⊢ Rel ∼ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcoss 38405 | . 2 ⊢ Rel ≀ (◡ E ↾ 𝐴) | |
2 | df-coels 38394 | . . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴) | |
3 | 2 | releqi 5790 | . 2 ⊢ (Rel ∼ 𝐴 ↔ Rel ≀ (◡ E ↾ 𝐴)) |
4 | 1, 3 | mpbir 231 | 1 ⊢ Rel ∼ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: E cep 5588 ◡ccnv 5688 ↾ cres 5691 Rel wrel 5694 ≀ ccoss 38162 ∼ ccoels 38163 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-ss 3980 df-opab 5211 df-xp 5695 df-rel 5696 df-coss 38393 df-coels 38394 |
This theorem is referenced by: erimeq2 38660 |
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