Theorem rnresequniqs 38328

Description: The range of a restriction is equal to the union of the quotient set. (Contributed by Peter Mazsa, 19-May-2018.)

Assertion

Ref	Expression
rnresequniqs	⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ran (𝑅 ↾ 𝐴) = ∪ (𝐴 / 𝑅))

Proof of Theorem rnresequniqs

Step	Hyp	Ref	Expression
1		uniqsALTV 38325	. 2 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴))
2		df-ima 5706	. 2 ⊢ (𝑅 “ 𝐴) = ran (𝑅 ↾ 𝐴)
3	1, 2	eqtr2di 2794	1 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ran (𝑅 ↾ 𝐴) = ∪ (𝐴 / 𝑅))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  ∪ cuni 4915  ran crn 5694   ↾ cres 5695   “ cima 5696   / cqs 8752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-iun 5001  df-br 5152  df-opab 5214  df-xp 5699  df-rel 5700  df-cnv 5701  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-ec 8755  df-qs 8759

This theorem is referenced by:  unidmqs  38650