Theorem uniqs2 8643

Description: The union of a quotient set. (Contributed by Mario Carneiro, 11-Jul-2014.)

Hypotheses

Ref	Expression
qsss.1	⊢ (𝜑 → 𝑅 Er 𝐴)
qsss.2	⊢ (𝜑 → 𝑅 ∈ 𝑉)

Assertion

Ref	Expression
uniqs2	⊢ (𝜑 → ∪ (𝐴 / 𝑅) = 𝐴)

Proof of Theorem uniqs2

Step	Hyp	Ref	Expression
1		qsss.2	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
2		uniqs 8641	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴))
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴))
4		qsss.1	. . . . . 6 ⊢ (𝜑 → 𝑅 Er 𝐴)
5		erdm 8583	. . . . . 6 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
6	4, 5	syl 17	. . . . 5 ⊢ (𝜑 → dom 𝑅 = 𝐴)
7	6	imaeq2d 6003	. . . 4 ⊢ (𝜑 → (𝑅 “ dom 𝑅) = (𝑅 “ 𝐴))
8	3, 7	eqtr4d 2780	. . 3 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = (𝑅 “ dom 𝑅))
9		imadmrn 6013	. . 3 ⊢ (𝑅 “ dom 𝑅) = ran 𝑅
10	8, 9	eqtrdi 2793	. 2 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = ran 𝑅)
11		errn 8595	. . 3 ⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)
12	4, 11	syl 17	. 2 ⊢ (𝜑 → ran 𝑅 = 𝐴)
13	10, 12	eqtrd 2777	1 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = 𝐴)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  ∪ cuni 4856  dom cdm 5624  ran crn 5625   “ cima 5627   Er wer 8570   / cqs 8572

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2708  ax-sep 5247  ax-nul 5254  ax-pr 5376  ax-un 7654

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3444  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4274  df-if 4478  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-iun 4947  df-br 5097  df-opab 5159  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-er 8573  df-ec 8575  df-qs 8579

This theorem is referenced by:  qshash  15638  cldsubg  23367  pi1buni  24308  qustrivr  31855