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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 38424 | . 2 ⊢ Rel ≀ (◡ E ↾ 𝐴) | |
| 2 | df-coels 38413 | . . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴) | |
| 3 | 2 | releqi 5787 | . 2 ⊢ (Rel ∼ 𝐴 ↔ Rel ≀ (◡ E ↾ 𝐴)) | 
| 4 | 1, 3 | mpbir 231 | 1 ⊢ Rel ∼ 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: E cep 5583 ◡ccnv 5684 ↾ cres 5687 Rel wrel 5690 ≀ ccoss 38182 ∼ ccoels 38183 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-opab 5206 df-xp 5691 df-rel 5692 df-coss 38412 df-coels 38413 | 
| This theorem is referenced by: erimeq2 38679 | 
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