Theorem uniqsw 8748

Description: The union of a quotient set. More restrictive antecedent; kept for backward compatibility; for new work, prefer uniqs 8747. (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 5998	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾ 𝐴) ∈ V)
2		uniqs 8747	. 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 1540   ∈ wcel 2109  Vcvv 3447  ∪ cuni 4871   ↾ cres 5640   “ cima 5641   / cqs 8670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ec 8673  df-qs 8677

This theorem is referenced by:  uniqs2  8750  ecqs  8752