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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 35651). (Contributed by Peter Mazsa, 27-Dec-2018.) |
Ref | Expression |
---|---|
dfcoss3 | ⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brcnvg 5743 | . . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥)) | |
2 | 1 | el2v 3500 | . . . . 5 ⊢ (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥) |
3 | 2 | anbi1i 625 | . . . 4 ⊢ ((𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦) ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) |
4 | 3 | exbii 1842 | . . 3 ⊢ (∃𝑢(𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦) ↔ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) |
5 | 4 | opabbii 5124 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦)} = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} |
6 | df-co 5557 | . 2 ⊢ (𝑅 ∘ ◡𝑅) = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦)} | |
7 | df-coss 35651 | . 2 ⊢ ≀ 𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} | |
8 | 5, 6, 7 | 3eqtr4ri 2853 | 1 ⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1531 ∃wex 1774 Vcvv 3493 class class class wbr 5057 {copab 5119 ◡ccnv 5547 ∘ ccom 5552 ≀ ccoss 35445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-rab 3145 df-v 3495 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-br 5058 df-opab 5120 df-cnv 5556 df-co 5557 df-coss 35651 |
This theorem is referenced by: cossex 35656 dmcoss3 35685 funALTVfun 35923 |
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