Theorem dmqseqim 39275

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 4884	. . 3 ⊢ ((dom 𝑅 / 𝑅) = 𝐴 → ∪ (dom 𝑅 / 𝑅) = ∪ 𝐴)
2		unidmqseq 39274	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → (∪ (dom 𝑅 / 𝑅) = ∪ 𝐴 ↔ ran 𝑅 = ∪ 𝐴)))
3	2	imp 411	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (∪ (dom 𝑅 / 𝑅) = ∪ 𝐴 ↔ ran 𝑅 = ∪ 𝐴))
4	1, 3	imbitrid 247	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → ((dom 𝑅 / 𝑅) = 𝐴 → ran 𝑅 = ∪ 𝐴))
5	4	ex 417	1 ⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → ran 𝑅 = ∪ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∪ cuni 4873  dom cdm 5659  ran crn 5660  Rel wrel 5664   / cqs 8689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-ext 2741  ax-sep 5258  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ec 8692  df-qs 8696

This theorem is referenced by:  dmqseqim2  39276