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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 5969 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 37830 | . . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦)) | |
2 | 1 | el2v 3477 | . . . 4 ⊢ (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦) |
3 | 2 | exbii 1843 | . . 3 ⊢ (∃𝑦 𝑦 ≀ 𝑅𝑥 ↔ ∃𝑦 𝑥 ≀ 𝑅𝑦) |
4 | 3 | abbii 2797 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦 ≀ 𝑅𝑥} = {𝑥 ∣ ∃𝑦 𝑥 ≀ 𝑅𝑦} |
5 | dfrn2 5885 | . 2 ⊢ ran ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑦 ≀ 𝑅𝑥} | |
6 | df-dm 5682 | . 2 ⊢ dom ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑥 ≀ 𝑅𝑦} | |
7 | 4, 5, 6 | 3eqtr4i 2765 | 1 ⊢ ran ≀ 𝑅 = dom ≀ 𝑅 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 ∃wex 1774 {cab 2704 Vcvv 3469 class class class wbr 5142 dom cdm 5672 ran crn 5673 ≀ ccoss 37570 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-cnv 5680 df-dm 5682 df-rn 5683 df-coss 37807 |
This theorem is referenced by: refrelcoss3 37859 |
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