Theorem unidmqs 38619

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 5987	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾ dom 𝑅) ∈ V)
2		rnresequniqs 38289	. . . 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 5986	. . . 4 ⊢ (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
5	4	rneqd 5891	. . 3 ⊢ (Rel 𝑅 → ran (𝑅 ↾ dom 𝑅) = ran 𝑅)
6	3, 5	sylan9req 2785	. 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 1540   ∈ wcel 2109  Vcvv 3444  ∪ cuni 4867  dom cdm 5631  ran crn 5632   ↾ cres 5633  Rel wrel 5636   / cqs 8647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ec 8650  df-qs 8654

This theorem is referenced by:  unidmqseq  38620