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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 36546 | . 2 ⊢ Rel ≀ (◡ E ↾ 𝐴) | |
2 | df-coels 36538 | . . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴) | |
3 | 2 | releqi 5688 | . 2 ⊢ (Rel ∼ 𝐴 ↔ Rel ≀ (◡ E ↾ 𝐴)) |
4 | 1, 3 | mpbir 230 | 1 ⊢ Rel ∼ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: E cep 5494 ◡ccnv 5588 ↾ cres 5591 Rel wrel 5594 ≀ ccoss 36333 ∼ ccoels 36334 |
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-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3434 df-in 3894 df-ss 3904 df-opab 5137 df-xp 5595 df-rel 5596 df-coss 36537 df-coels 36538 |
This theorem is referenced by: erim2 36789 |
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