| Mathbox for Peter Mazsa |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > unidmqs | Structured version Visualization version GIF version | ||
| Description: The range of a relation is equal to the union of the domain quotient. (Contributed by Peter Mazsa, 13-Oct-2018.) |
| Ref | Expression |
|---|---|
| unidmqs | ⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ∪ (dom 𝑅 / 𝑅) = ran 𝑅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | resexg 6045 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾ dom 𝑅) ∈ V) | |
| 2 | rnresequniqs 38333 | . . . 4 ⊢ ((𝑅 ↾ dom 𝑅) ∈ V → ran (𝑅 ↾ dom 𝑅) = ∪ (dom 𝑅 / 𝑅)) | |
| 3 | 1, 2 | syl 17 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ran (𝑅 ↾ dom 𝑅) = ∪ (dom 𝑅 / 𝑅)) |
| 4 | resdm 6044 | . . . 4 ⊢ (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅) | |
| 5 | 4 | rneqd 5949 | . . 3 ⊢ (Rel 𝑅 → ran (𝑅 ↾ dom 𝑅) = ran 𝑅) |
| 6 | 3, 5 | sylan9req 2798 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → ∪ (dom 𝑅 / 𝑅) = ran 𝑅) |
| 7 | 6 | ex 412 | 1 ⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ∪ (dom 𝑅 / 𝑅) = ran 𝑅)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cuni 4907 dom cdm 5685 ran crn 5686 ↾ cres 5687 Rel wrel 5690 / cqs 8744 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| 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-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ec 8747 df-qs 8751 |
| This theorem is referenced by: unidmqseq 38656 |
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