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Mathbox for Peter Mazsa |
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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 5970 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 37958 | . . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦)) | |
2 | 1 | el2v 3471 | . . . 4 ⊢ (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦) |
3 | 2 | exbii 1842 | . . 3 ⊢ (∃𝑦 𝑦 ≀ 𝑅𝑥 ↔ ∃𝑦 𝑥 ≀ 𝑅𝑦) |
4 | 3 | abbii 2795 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦 ≀ 𝑅𝑥} = {𝑥 ∣ ∃𝑦 𝑥 ≀ 𝑅𝑦} |
5 | dfrn2 5886 | . 2 ⊢ ran ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑦 ≀ 𝑅𝑥} | |
6 | df-dm 5683 | . 2 ⊢ dom ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑥 ≀ 𝑅𝑦} | |
7 | 4, 5, 6 | 3eqtr4i 2763 | 1 ⊢ ran ≀ 𝑅 = dom ≀ 𝑅 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∃wex 1773 {cab 2702 Vcvv 3463 class class class wbr 5144 dom cdm 5673 ran crn 5674 ≀ ccoss 37701 |
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 2696 ax-sep 5295 ax-nul 5302 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3420 df-v 3465 df-dif 3944 df-un 3946 df-ss 3958 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-br 5145 df-opab 5207 df-cnv 5681 df-dm 5683 df-rn 5684 df-coss 37935 |
This theorem is referenced by: refrelcoss3 37987 |
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