Theorem uniqsw 8795

Description: The union of a quotient set. More restrictive antecedent; kept for backward compatibility; for new work, prefer uniqs 8794. (Contributed by NM, 9-Dec-2008.) (Proof shortened by AV, 25-Nov-2025.)

Assertion

Ref	Expression
uniqsw	⊢ (𝑅 ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴))

Proof of Theorem uniqsw

Step	Hyp	Ref	Expression
1		resexg 6015	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾ 𝐴) ∈ V)
2		uniqs 8794	. 2 ⊢ ((𝑅 ↾ 𝐴) ∈ V → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴))
3	1, 2	syl 18	1 ⊢ (𝑅 ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ∪ cuni 4866   ↾ cres 5652   “ cima 5653   / cqs 8716

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-11 2193  ax-ext 2732  ax-sep 5248  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ec 8719  df-qs 8723

This theorem is used by:  uniqs2  8797  ecqs  8800