Theorem uniqs2 8779

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 8777	. . . . 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 8719	. . . . . 6 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
6	4, 5	syl 17	. . . . 5 ⊢ (𝜑 → dom 𝑅 = 𝐴)
7	6	imaeq2d 6059	. . . 4 ⊢ (𝜑 → (𝑅 “ dom 𝑅) = (𝑅 “ 𝐴))
8	3, 7	eqtr4d 2774	. . 3 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = (𝑅 “ dom 𝑅))
9		imadmrn 6069	. . 3 ⊢ (𝑅 “ dom 𝑅) = ran 𝑅
10	8, 9	eqtrdi 2787	. 2 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = ran 𝑅)
11		errn 8731	. . 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 2105  ∪ cuni 4908  dom cdm 5676  ran crn 5677   “ cima 5679   Er wer 8706   / cqs 8708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-11 2153  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-er 8709  df-ec 8711  df-qs 8715

This theorem is referenced by:  qshash  15780  cldsubg  23935  pi1buni  24887  qustrivr  32917