| Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > rncossdmcoss | Structured version Visualization version GIF version | ||
| Description: The range of cosets is the domain of them (this should be rncoss 5986 but there exists a theorem with this name already). (Contributed by Peter Mazsa, 12-Dec-2019.) |
| Ref | Expression |
|---|---|
| rncossdmcoss | ⊢ ran ≀ 𝑅 = dom ≀ 𝑅 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | brcosscnvcoss 38435 | . . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦)) | |
| 2 | 1 | el2v 3487 | . . . 4 ⊢ (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦) |
| 3 | 2 | exbii 1848 | . . 3 ⊢ (∃𝑦 𝑦 ≀ 𝑅𝑥 ↔ ∃𝑦 𝑥 ≀ 𝑅𝑦) |
| 4 | 3 | abbii 2809 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦 ≀ 𝑅𝑥} = {𝑥 ∣ ∃𝑦 𝑥 ≀ 𝑅𝑦} |
| 5 | dfrn2 5899 | . 2 ⊢ ran ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑦 ≀ 𝑅𝑥} | |
| 6 | df-dm 5695 | . 2 ⊢ dom ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑥 ≀ 𝑅𝑦} | |
| 7 | 4, 5, 6 | 3eqtr4i 2775 | 1 ⊢ ran ≀ 𝑅 = dom ≀ 𝑅 |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∃wex 1779 {cab 2714 Vcvv 3480 class class class wbr 5143 dom cdm 5685 ran crn 5686 ≀ ccoss 38182 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 df-coss 38412 |
| This theorem is referenced by: refrelcoss3 38464 |
| Copyright terms: Public domain | W3C validator |