Theorem uniqs 8796

Description: The union of a quotient set. (Contributed by NM, 9-Dec-2008.)

Assertion

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

Proof of Theorem uniqs

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ecexg 8729	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → [𝑥]𝑅 ∈ V)
2	1	ralrimivw 3139	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 [𝑥]𝑅 ∈ V)
3		dfiun2g 5034	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 [𝑥]𝑅 ∈ V → ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅})
4	2, 3	syl 17	. . 3 ⊢ (𝑅 ∈ 𝑉 → ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅})
5	4	eqcomd 2731	. 2 ⊢ (𝑅 ∈ 𝑉 → ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} = ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅)
6		df-qs 8731	. . 3 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
7	6	unieqi 4921	. 2 ⊢ ∪ (𝐴 / 𝑅) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
8		df-ec 8727	. . . . 5 ⊢ [𝑥]𝑅 = (𝑅 “ {𝑥})
9	8	a1i 11	. . . 4 ⊢ (𝑥 ∈ 𝐴 → [𝑥]𝑅 = (𝑅 “ {𝑥}))
10	9	iuneq2i 5018	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅 = ∪ 𝑥 ∈ 𝐴 (𝑅 “ {𝑥})
11		imaiun 7255	. . 3 ⊢ (𝑅 “ ∪ 𝑥 ∈ 𝐴 {𝑥}) = ∪ 𝑥 ∈ 𝐴 (𝑅 “ {𝑥})
12		iunid 5064	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
13	12	imaeq2i 6062	. . 3 ⊢ (𝑅 “ ∪ 𝑥 ∈ 𝐴 {𝑥}) = (𝑅 “ 𝐴)
14	10, 11, 13	3eqtr2ri 2760	. 2 ⊢ (𝑅 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅
15	5, 7, 14	3eqtr4g 2790	1 ⊢ (𝑅 ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  {cab 2702  ∀wral 3050  ∃wrex 3059  Vcvv 3461  {csn 4630  ∪ cuni 4909  ∪ ciun 4997   “ cima 5681  [cec 8723   / cqs 8724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ec 8727  df-qs 8731

This theorem is referenced by:  uniqs2  8798  ecqs  8800