Theorem uniqsw 8758

Description: The union of a quotient set. More restrictive antecedent; kept for backward compatibility; for new work, prefer uniqs 8757. (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 8757	. 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 1562   ∈ wcel 2144  Vcvv 3456  ∪ cuni 4867   ↾ cres 5651   “ cima 5652   / cqs 8679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-11 2193  ax-ext 2736  ax-sep 5248  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-xp 5655  df-rel 5656  df-cnv 5657  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ec 8682  df-qs 8686

This theorem is referenced by:  uniqs2  8760  ecqs  8763