Theorem unidmqs 38825

Description: The range of a relation is equal to the union of the domain quotient. (Contributed by Peter Mazsa, 13-Oct-2018.)

Assertion

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

Proof of Theorem unidmqs

Step	Hyp	Ref	Expression
1		resexg 5983	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾ dom 𝑅) ∈ V)
2		rnresequniqs 38439	. . . 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 5982	. . . 4 ⊢ (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
5	4	rneqd 5884	. . 3 ⊢ (Rel 𝑅 → ran (𝑅 ↾ dom 𝑅) = ran 𝑅)
6	3, 5	sylan9req 2789	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → ∪ (dom 𝑅 / 𝑅) = ran 𝑅)
7	6	ex 412	1 ⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ∪ (dom 𝑅 / 𝑅) = ran 𝑅))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3437  ∪ cuni 4860  dom cdm 5621  ran crn 5622   ↾ cres 5623  Rel wrel 5626   / cqs 8630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-xp 5627  df-rel 5628  df-cnv 5629  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ec 8633  df-qs 8637

This theorem is referenced by:  unidmqseq  38826