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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfcoss3 | Structured version Visualization version GIF version |
Description: Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 37737). (Contributed by Peter Mazsa, 27-Dec-2018.) |
Ref | Expression |
---|---|
dfcoss3 | ⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brcnvg 5869 | . . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥)) | |
2 | 1 | el2v 3474 | . . . . 5 ⊢ (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥) |
3 | 2 | anbi1i 623 | . . . 4 ⊢ ((𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦) ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) |
4 | 3 | exbii 1842 | . . 3 ⊢ (∃𝑢(𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦) ↔ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) |
5 | 4 | opabbii 5205 | . 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} |
6 | df-co 5675 | . 2 ⊢ (𝑅 ∘ ◡𝑅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦)} | |
7 | df-coss 37737 | . 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} | |
8 | 5, 6, 7 | 3eqtr4ri 2763 | 1 ⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1533 ∃wex 1773 Vcvv 3466 class class class wbr 5138 {copab 5200 ◡ccnv 5665 ∘ ccom 5670 ≀ ccoss 37499 |
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 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-br 5139 df-opab 5201 df-cnv 5674 df-co 5675 df-coss 37737 |
This theorem is referenced by: cossex 37745 dmcoss3 37779 funALTVfun 38024 |
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