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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 6028 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾ dom 𝑅) ∈ V) | |
2 | rnresequniqs 38043 | . . . 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 6027 | . . . 4 ⊢ (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅) | |
5 | 4 | rneqd 5936 | . . 3 ⊢ (Rel 𝑅 → ran (𝑅 ↾ dom 𝑅) = ran 𝑅) |
6 | 3, 5 | sylan9req 2787 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → ∪ (dom 𝑅 / 𝑅) = ran 𝑅) |
7 | 6 | ex 411 | 1 ⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ∪ (dom 𝑅 / 𝑅) = ran 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ∪ cuni 4905 dom cdm 5674 ran crn 5675 ↾ cres 5676 Rel wrel 5679 / cqs 8725 |
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-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-xp 5680 df-rel 5681 df-cnv 5682 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-ec 8728 df-qs 8732 |
This theorem is referenced by: unidmqseq 38366 |
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