Theorem qsex 3231

Description: A quotient set exists.

Hypothesis

Ref	Expression
qsex.1	⊢ A ∈ V

Assertion

Ref	Expression
qsex	⊢ (A / R) ∈ V

Proof of Theorem qsex

Step	Hyp	Ref	Expression
1		df-rex 1206	. . . 4 ⊢ (∃x ∈ A y = [x]R ↔ ∃x(x ∈ A ∧ y = [x]R))
2	1	biabi 1181	. . 3 ⊢ {y∣∃x ∈ A y = [x]R} = {y∣∃x(x ∈ A ∧ y = [x]R)}
3		df-qs 3205	. . 3 ⊢ (A / R) = {y∣∃x ∈ A y = [x]R}
4		rnopab 2566	. . 3 ⊢ ran {⟨x, y⟩∣(x ∈ A ∧ y = [x]R)} = {y∣∃x(x ∈ A ∧ y = [x]R)}
5	2, 3, 4	3eqtr4 1126	. 2 ⊢ (A / R) = ran {⟨x, y⟩∣(x ∈ A ∧ y = [x]R)}
6		qsex.1	. . . 4 ⊢ A ∈ V
7		dmopabss 2540	. . . 4 ⊢ dom {⟨x, y⟩∣(x ∈ A ∧ y = [x]R)} ⊆ A
8	6, 7	ssexi 1701	. . 3 ⊢ dom {⟨x, y⟩∣(x ∈ A ∧ y = [x]R)} ∈ V
9		funopab 2694	. . . 4 ⊢ (Fun {⟨x, y⟩∣(x ∈ A ∧ y = [x]R)} ↔ ∀x∃*y(x ∈ A ∧ y = [x]R))
10		moeq 1431	. . . . 5 ⊢ ∃*y y = [x]R
11	10	moani 1047	. . . 4 ⊢ ∃*y(x ∈ A ∧ y = [x]R)
12	9, 11	mpgbir 686	. . 3 ⊢ Fun {⟨x, y⟩∣(x ∈ A ∧ y = [x]R)}
13		funrnex 2743	. . 3 ⊢ (dom {⟨x, y⟩∣(x ∈ A ∧ y = [x]R)} ∈ V → (Fun {⟨x, y⟩∣(x ∈ A ∧ y = [x]R)} → ran {⟨x, y⟩∣(x ∈ A ∧ y = [x]R)} ∈ V))
14	8, 12, 13	mp2 43	. 2 ⊢ ran {⟨x, y⟩∣(x ∈ A ∧ y = [x]R)} ∈ V
15	5, 14	eqeltr 1159	1 ⊢ (A / R) ∈ V

Syntax hints:   ∧ wa 196  ∃wex 678  ∃*wmo 1008  {cab 1090   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  Vcvv 1348  {copab 2055  dom cdm 2410  ran crn 2411  Fun wfun 2416  [cec 3198   / cqs 3199

This theorem is referenced by:  nqex 3843  srex 3973

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-qs 3205

metamath.org