Theorem uniqs 8773

Description: The union of a quotient set, like uniqsw 8774 but with a weaker antecedent: only the restriction of 𝑅 by 𝐴 needs to be a set, not 𝑅 itself, see e.g. cnvepima 39085. (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 8747	. . . . 5 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → (𝑥 ∈ 𝐴 → [𝑥]𝑅 ∈ V))
2	1	ralrimiv 3153	. . . 4 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ∀𝑥 ∈ 𝐴 [𝑥]𝑅 ∈ V)
3		dfiun2g 4988	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 [𝑥]𝑅 ∈ V → ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅})
4	2, 3	syl 18	. . 3 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅})
5	4	eqcomd 2766	. 2 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} = ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅)
6		df-qs 8702	. . 3 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
7	6	unieqi 4879	. 2 ⊢ ∪ (𝐴 / 𝑅) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
8		df-ec 8698	. . . . 5 ⊢ [𝑥]𝑅 = (𝑅 “ {𝑥})
9	8	a1i 11	. . . 4 ⊢ (𝑥 ∈ 𝐴 → [𝑥]𝑅 = (𝑅 “ {𝑥}))
10	9	iuneq2i 4973	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅 = ∪ 𝑥 ∈ 𝐴 (𝑅 “ {𝑥})
11		imaiun 7242	. . 3 ⊢ (𝑅 “ ∪ 𝑥 ∈ 𝐴 {𝑥}) = ∪ 𝑥 ∈ 𝐴 (𝑅 “ {𝑥})
12		iunid 5019	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
13	12	imaeq2i 6054	. . 3 ⊢ (𝑅 “ ∪ 𝑥 ∈ 𝐴 {𝑥}) = (𝑅 “ 𝐴)
14	10, 11, 13	3eqtr2ri 2790	. 2 ⊢ (𝑅 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅
15	5, 7, 14	3eqtr4g 2820	1 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076  ∃wrex 3086  Vcvv 3450  {csn 4584  ∪ cuni 4867  ∪ ciun 4951   ↾ cres 5657   “ cima 5658  [cec 8694   / cqs 8695

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2732  ax-sep 5251  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ec 8698  df-qs 8702

This theorem is used by:  uniqsw  8774  rnresequniqs  39082  cnvepima  39085