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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 37269). (Contributed by Peter Mazsa, 27-Dec-2018.) |
Ref | Expression |
---|---|
dfcoss3 | ⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brcnvg 5877 | . . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥)) | |
2 | 1 | el2v 3482 | . . . . 5 ⊢ (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥) |
3 | 2 | anbi1i 624 | . . . 4 ⊢ ((𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦) ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) |
4 | 3 | exbii 1850 | . . 3 ⊢ (∃𝑢(𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦) ↔ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) |
5 | 4 | opabbii 5214 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦)} = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} |
6 | df-co 5684 | . 2 ⊢ (𝑅 ∘ ◡𝑅) = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦)} | |
7 | df-coss 37269 | . 2 ⊢ ≀ 𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} | |
8 | 5, 6, 7 | 3eqtr4ri 2771 | 1 ⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∃wex 1781 Vcvv 3474 class class class wbr 5147 {copab 5209 ◡ccnv 5674 ∘ ccom 5679 ≀ ccoss 37031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-cnv 5683 df-co 5684 df-coss 37269 |
This theorem is referenced by: cossex 37277 dmcoss3 37311 funALTVfun 37556 |
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