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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 35674). (Contributed by Peter Mazsa, 27-Dec-2018.) |
Ref | Expression |
---|---|
dfcoss3 | ⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brcnvg 5750 | . . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥)) | |
2 | 1 | el2v 3501 | . . . . 5 ⊢ (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥) |
3 | 2 | anbi1i 625 | . . . 4 ⊢ ((𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦) ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) |
4 | 3 | exbii 1848 | . . 3 ⊢ (∃𝑢(𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦) ↔ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) |
5 | 4 | opabbii 5133 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦)} = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} |
6 | df-co 5564 | . 2 ⊢ (𝑅 ∘ ◡𝑅) = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦)} | |
7 | df-coss 35674 | . 2 ⊢ ≀ 𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} | |
8 | 5, 6, 7 | 3eqtr4ri 2855 | 1 ⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 Vcvv 3494 class class class wbr 5066 {copab 5128 ◡ccnv 5554 ∘ ccom 5559 ≀ ccoss 35468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-cnv 5563 df-co 5564 df-coss 35674 |
This theorem is referenced by: cossex 35679 dmcoss3 35708 funALTVfun 35946 |
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