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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 5841 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 36294 | . . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦)) | |
2 | 1 | el2v 3416 | . . . 4 ⊢ (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦) |
3 | 2 | exbii 1855 | . . 3 ⊢ (∃𝑦 𝑦 ≀ 𝑅𝑥 ↔ ∃𝑦 𝑥 ≀ 𝑅𝑦) |
4 | 3 | abbii 2808 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦 ≀ 𝑅𝑥} = {𝑥 ∣ ∃𝑦 𝑥 ≀ 𝑅𝑦} |
5 | dfrn2 5757 | . 2 ⊢ ran ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑦 ≀ 𝑅𝑥} | |
6 | df-dm 5561 | . 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 209 = wceq 1543 ∃wex 1787 {cab 2714 Vcvv 3408 class class class wbr 5053 dom cdm 5551 ran crn 5552 ≀ ccoss 36070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-cnv 5559 df-dm 5561 df-rn 5562 df-coss 36274 |
This theorem is referenced by: refrelcoss3 36318 |
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