Theorem uniqs2 8798

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 8796	. . . . 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 8734	. . . . . 6 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
6	4, 5	syl 17	. . . . 5 ⊢ (𝜑 → dom 𝑅 = 𝐴)
7	6	imaeq2d 6052	. . . 4 ⊢ (𝜑 → (𝑅 “ dom 𝑅) = (𝑅 “ 𝐴))
8	3, 7	eqtr4d 2774	. . 3 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = (𝑅 “ dom 𝑅))
9		imadmrn 6062	. . 3 ⊢ (𝑅 “ dom 𝑅) = ran 𝑅
10	8, 9	eqtrdi 2787	. 2 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = ran 𝑅)
11		errn 8746	. . 3 ⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)
12	4, 11	syl 17	. 2 ⊢ (𝜑 → ran 𝑅 = 𝐴)
13	10, 12	eqtrd 2771	1 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = 𝐴)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∪ cuni 4888  dom cdm 5659  ran crn 5660   “ cima 5662   Er wer 8721   / cqs 8723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-er 8724  df-ec 8726  df-qs 8730

This theorem is referenced by:  qshash  15848  cldsubg  24054  pi1buni  24996  qustrivr  33385