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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 37597 | . 2 ⊢ Rel ≀ (◡ E ↾ 𝐴) | |
2 | df-coels 37586 | . . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴) | |
3 | 2 | releqi 5777 | . 2 ⊢ (Rel ∼ 𝐴 ↔ Rel ≀ (◡ E ↾ 𝐴)) |
4 | 1, 3 | mpbir 230 | 1 ⊢ Rel ∼ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: E cep 5579 ◡ccnv 5675 ↾ cres 5678 Rel wrel 5681 ≀ ccoss 37347 ∼ ccoels 37348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3475 df-in 3955 df-ss 3965 df-opab 5211 df-xp 5682 df-rel 5683 df-coss 37585 df-coels 37586 |
This theorem is referenced by: erimeq2 37852 |
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