Theorem uniqsw 8716

Description: The union of a quotient set. More restrictive antecedent; kept for backward compatibility; for new work, prefer uniqs 8715. (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 5988	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾ 𝐴) ∈ V)
2		uniqs 8715	. 2 ⊢ ((𝑅 ↾ 𝐴) ∈ V → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴))
3	1, 2	syl 17	1 ⊢ (𝑅 ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ∪ cuni 4851   ↾ cres 5628   “ cima 5629   / cqs 8637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709  ax-sep 5232  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-xp 5632  df-rel 5633  df-cnv 5634  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-ec 8640  df-qs 8644

This theorem is referenced by:  uniqs2  8718  ecqs  8721