Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > dmqseqim | Structured version Visualization version GIF version |
Description: If the domain quotient of a relation is equal to the class 𝐴, then the range of the relation is the union of the class. (Contributed by Peter Mazsa, 29-Dec-2021.) |
Ref | Expression |
---|---|
dmqseqim | ⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → ran 𝑅 = ∪ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unieq 4830 | . . 3 ⊢ ((dom 𝑅 / 𝑅) = 𝐴 → ∪ (dom 𝑅 / 𝑅) = ∪ 𝐴) | |
2 | unidmqseq 36504 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → (∪ (dom 𝑅 / 𝑅) = ∪ 𝐴 ↔ ran 𝑅 = ∪ 𝐴))) | |
3 | 2 | imp 410 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (∪ (dom 𝑅 / 𝑅) = ∪ 𝐴 ↔ ran 𝑅 = ∪ 𝐴)) |
4 | 1, 3 | syl5ib 247 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → ((dom 𝑅 / 𝑅) = 𝐴 → ran 𝑅 = ∪ 𝐴)) |
5 | 4 | ex 416 | 1 ⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → ran 𝑅 = ∪ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∪ cuni 4819 dom cdm 5551 ran crn 5552 Rel wrel 5556 / cqs 8390 |
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-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
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-nf 1792 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-xp 5557 df-rel 5558 df-cnv 5559 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-ec 8393 df-qs 8397 |
This theorem is referenced by: dmqseqim2 36506 |
Copyright terms: Public domain | W3C validator |