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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 35548 | . 2 ⊢ Rel ≀ (◡ E ↾ 𝐴) | |
2 | df-coels 35540 | . . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴) | |
3 | 2 | releqi 5645 | . 2 ⊢ (Rel ∼ 𝐴 ↔ Rel ≀ (◡ E ↾ 𝐴)) |
4 | 1, 3 | mpbir 232 | 1 ⊢ Rel ∼ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: E cep 5457 ◡ccnv 5547 ↾ cres 5550 Rel wrel 5553 ≀ ccoss 35334 ∼ ccoels 35335 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-opab 5120 df-xp 5554 df-rel 5555 df-coss 35539 df-coels 35540 |
This theorem is referenced by: erim2 35791 |
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