Theorem unidmqseq 36767

Description: The union of the domain quotient of a relation is equal to the class 𝐴 if and only if the range is equal to it as well. (Contributed by Peter Mazsa, 21-Apr-2019.) (Revised by Peter Mazsa, 28-Dec-2021.)

Assertion

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

Proof of Theorem unidmqseq

Step	Hyp	Ref	Expression
1		unidmqs 36766	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ∪ (dom 𝑅 / 𝑅) = ran 𝑅))
2	1	imp 407	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → ∪ (dom 𝑅 / 𝑅) = ran 𝑅)
3	2	eqeq1d 2740	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (∪ (dom 𝑅 / 𝑅) = 𝐴 ↔ ran 𝑅 = 𝐴))
4	3	ex 413	1 ⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → (∪ (dom 𝑅 / 𝑅) = 𝐴 ↔ ran 𝑅 = 𝐴)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∪ cuni 4839  dom cdm 5589  ran crn 5590  Rel wrel 5594   / cqs 8497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ec 8500  df-qs 8504

This theorem is referenced by:  dmqseqim  36768