Theorem dmqseqim 36505

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.)

Assertion

Ref	Expression
dmqseqim	⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → ran 𝑅 = ∪ 𝐴)))

Proof of Theorem dmqseqim

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)))

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