Theorem uniqs 8722

Description: The union of a quotient set, like uniqsw 8723 but with a weaker antecedent: only the restriction of 𝑅 by 𝐴 needs to be a set, not 𝑅 itself, see e.g. cnvepima 38588. (Contributed by NM, 9-Dec-2008.) (Revised by Peter Mazsa, 20-Jun-2019.)

Assertion

Ref	Expression
uniqs	⊢ ((𝑅 ↾ 𝐴) ∈ 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		elecex 8696	. . . . 5 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → (𝑥 ∈ 𝐴 → [𝑥]𝑅 ∈ V))
2	1	ralrimiv 3129	. . . 4 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ∀𝑥 ∈ 𝐴 [𝑥]𝑅 ∈ V)
3		dfiun2g 4987	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 [𝑥]𝑅 ∈ V → ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅})
4	2, 3	syl 17	. . 3 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅})
5	4	eqcomd 2743	. 2 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} = ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅)
6		df-qs 8651	. . 3 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
7	6	unieqi 4877	. 2 ⊢ ∪ (𝐴 / 𝑅) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
8		df-ec 8647	. . . . 5 ⊢ [𝑥]𝑅 = (𝑅 “ {𝑥})
9	8	a1i 11	. . . 4 ⊢ (𝑥 ∈ 𝐴 → [𝑥]𝑅 = (𝑅 “ {𝑥}))
10	9	iuneq2i 4970	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅 = ∪ 𝑥 ∈ 𝐴 (𝑅 “ {𝑥})
11		imaiun 7201	. . 3 ⊢ (𝑅 “ ∪ 𝑥 ∈ 𝐴 {𝑥}) = ∪ 𝑥 ∈ 𝐴 (𝑅 “ {𝑥})
12		iunid 5018	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
13	12	imaeq2i 6025	. . 3 ⊢ (𝑅 “ ∪ 𝑥 ∈ 𝐴 {𝑥}) = (𝑅 “ 𝐴)
14	10, 11, 13	3eqtr2ri 2767	. 2 ⊢ (𝑅 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅
15	5, 7, 14	3eqtr4g 2797	1 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062  Vcvv 3442  {csn 4582  ∪ cuni 4865  ∪ ciun 4948   ↾ cres 5634   “ cima 5635  [cec 8643   / cqs 8644

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 5243  ax-pr 5379  ax-un 7690

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 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ec 8647  df-qs 8651

This theorem is referenced by:  uniqsw  8723  rnresequniqs  38585  cnvepima  38588