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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 5928 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 36942 | . . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦)) | |
2 | 1 | el2v 3452 | . . . 4 ⊢ (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦) |
3 | 2 | exbii 1851 | . . 3 ⊢ (∃𝑦 𝑦 ≀ 𝑅𝑥 ↔ ∃𝑦 𝑥 ≀ 𝑅𝑦) |
4 | 3 | abbii 2803 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦 ≀ 𝑅𝑥} = {𝑥 ∣ ∃𝑦 𝑥 ≀ 𝑅𝑦} |
5 | dfrn2 5845 | . 2 ⊢ ran ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑦 ≀ 𝑅𝑥} | |
6 | df-dm 5644 | . 2 ⊢ dom ≀ 𝑅 = {𝑥 ∣ ∃𝑦 𝑥 ≀ 𝑅𝑦} | |
7 | 4, 5, 6 | 3eqtr4i 2771 | 1 ⊢ ran ≀ 𝑅 = dom ≀ 𝑅 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ∃wex 1782 {cab 2710 Vcvv 3444 class class class wbr 5106 dom cdm 5634 ran crn 5635 ≀ ccoss 36680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-cnv 5642 df-dm 5644 df-rn 5645 df-coss 36919 |
This theorem is referenced by: refrelcoss3 36971 |
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