Theorem rnresequniqs 38794

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		uniqs 8749	. 2 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴))
2		df-ima 5656	. 2 ⊢ (𝑅 “ 𝐴) = ran (𝑅 ↾ 𝐴)
3	1, 2	eqtr2di 2813	1 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ran (𝑅 ↾ 𝐴) = ∪ (𝐴 / 𝑅))

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  ∪ cuni 4862  ran crn 5644   ↾ cres 5645   “ cima 5646   / cqs 8671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-11 2190  ax-ext 2733  ax-sep 5243  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-xp 5649  df-rel 5650  df-cnv 5651  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-ec 8674  df-qs 8678

This theorem is referenced by:  unidmqs  39199