Theorem rnresequniqs 38362

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 8693	. 2 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴))
2		df-ima 5624	. 2 ⊢ (𝑅 “ 𝐴) = ran (𝑅 ↾ 𝐴)
3	1, 2	eqtr2di 2783	1 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ran (𝑅 ↾ 𝐴) = ∪ (𝐴 / 𝑅))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ∪ cuni 4854  ran crn 5612   ↾ cres 5613   “ cima 5614   / cqs 8616

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 2113  ax-9 2121  ax-11 2160  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

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 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-xp 5617  df-rel 5618  df-cnv 5619  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-ec 8619  df-qs 8623

This theorem is referenced by:  unidmqs  38692