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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 37935 | . 2 ⊢ Rel ≀ (◡ E ↾ 𝐴) | |
2 | df-coels 37924 | . . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴) | |
3 | 2 | releqi 5783 | . 2 ⊢ (Rel ∼ 𝐴 ↔ Rel ≀ (◡ E ↾ 𝐴)) |
4 | 1, 3 | mpbir 230 | 1 ⊢ Rel ∼ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: E cep 5585 ◡ccnv 5681 ↾ cres 5684 Rel wrel 5687 ≀ ccoss 37689 ∼ ccoels 37690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3475 df-in 3956 df-ss 3966 df-opab 5215 df-xp 5688 df-rel 5689 df-coss 37923 df-coels 37924 |
This theorem is referenced by: erimeq2 38190 |
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