Theorem uniqs 5985

Description: The union of a quotient set. (Contributed by set.mm contributors, 9-Dec-2008.)

Assertion

Ref	Expression
uniqs	⊢ (R ∈ 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		ecexg 5950	. . . . 5 ⊢ (R ∈ V → [x]R ∈ V)
2	1	ralrimivw 2699	. . . 4 ⊢ (R ∈ V → ∀x ∈ A [x]R ∈ V)
3		dfiun2g 4000	. . . 4 ⊢ (∀x ∈ A [x]R ∈ V → ∪x ∈ A [x]R = ∪{y ∣ ∃x ∈ A y = [x]R})
4	2, 3	syl 15	. . 3 ⊢ (R ∈ V → ∪x ∈ A [x]R = ∪{y ∣ ∃x ∈ A y = [x]R})
5	4	eqcomd 2358	. 2 ⊢ (R ∈ V → ∪{y ∣ ∃x ∈ A y = [x]R} = ∪x ∈ A [x]R)
6		df-qs 5952	. . 3 ⊢ (A / R) = {y ∣ ∃x ∈ A y = [x]R}
7	6	unieqi 3902	. 2 ⊢ ∪(A / R) = ∪{y ∣ ∃x ∈ A y = [x]R}
8		df-ec 5948	. . . . 5 ⊢ [x]R = (R “ {x})
9	8	a1i 10	. . . 4 ⊢ (x ∈ A → [x]R = (R “ {x}))
10	9	iuneq2i 3988	. . 3 ⊢ ∪x ∈ A [x]R = ∪x ∈ A (R “ {x})
11		imaiun 5465	. . 3 ⊢ (R “ ∪x ∈ A {x}) = ∪x ∈ A (R “ {x})
12		iunid 4022	. . . 4 ⊢ ∪x ∈ A {x} = A
13	12	imaeq2i 4941	. . 3 ⊢ (R “ ∪x ∈ A {x}) = (R “ A)
14	10, 11, 13	3eqtr2ri 2380	. 2 ⊢ (R “ A) = ∪x ∈ A [x]R
15	5, 7, 14	3eqtr4g 2410	1 ⊢ (R ∈ V → ∪(A / R) = (R “ A))

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2615  ∃wrex 2616  Vcvv 2860  {csn 3738  ∪cuni 3892  ∪ciun 3970   “ cima 4723  [cec 5946   / cqs 5947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-br 4641  df-ima 4728  df-ec 5948  df-qs 5952

This theorem is referenced by:  uniqs2  5986  ecqs  5989