Theorem quslem 12744

Description: The function in qusval 12743 is a surjection onto a quotient set. (Contributed by Mario Carneiro, 23-Feb-2015.)

Hypotheses

Ref	Expression
qusval.u	⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))
qusval.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
qusval.f	⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
qusval.e	⊢ (𝜑 → ∼ ∈ 𝑊)
qusval.r	⊢ (𝜑 → 𝑅 ∈ 𝑍)

Assertion

Ref	Expression
quslem	⊢ (𝜑 → 𝐹:𝑉–onto→(𝑉 / ∼ ))

Distinct variable groups:   𝑥, ∼   𝜑,𝑥   𝑥,𝑅   𝑥,𝑉

Allowed substitution hints:   𝑈(𝑥)   𝐹(𝑥)   𝑊(𝑥)   𝑍(𝑥)

Proof of Theorem quslem

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		qusval.e	. . . . . 6 ⊢ (𝜑 → ∼ ∈ 𝑊)
2		ecexg 6538	. . . . . 6 ⊢ ( ∼ ∈ 𝑊 → [𝑥] ∼ ∈ V)
3	1, 2	syl 14	. . . . 5 ⊢ (𝜑 → [𝑥] ∼ ∈ V)
4	3	ralrimivw 2551	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V)
5		qusval.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
6	5	fnmpt 5342	. . . 4 ⊢ (∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V → 𝐹 Fn 𝑉)
7	4, 6	syl 14	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝑉)
8		dffn4 5444	. . 3 ⊢ (𝐹 Fn 𝑉 ↔ 𝐹:𝑉–onto→ran 𝐹)
9	7, 8	sylib 122	. 2 ⊢ (𝜑 → 𝐹:𝑉–onto→ran 𝐹)
10	5	rnmpt 4875	. . . 4 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝑉 𝑦 = [𝑥] ∼ }
11		df-qs 6540	. . . 4 ⊢ (𝑉 / ∼ ) = {𝑦 ∣ ∃𝑥 ∈ 𝑉 𝑦 = [𝑥] ∼ }
12	10, 11	eqtr4i 2201	. . 3 ⊢ ran 𝐹 = (𝑉 / ∼ )
13		foeq3 5436	. . 3 ⊢ (ran 𝐹 = (𝑉 / ∼ ) → (𝐹:𝑉–onto→ran 𝐹 ↔ 𝐹:𝑉–onto→(𝑉 / ∼ )))
14	12, 13	ax-mp 5	. 2 ⊢ (𝐹:𝑉–onto→ran 𝐹 ↔ 𝐹:𝑉–onto→(𝑉 / ∼ ))
15	9, 14	sylib 122	1 ⊢ (𝜑 → 𝐹:𝑉–onto→(𝑉 / ∼ ))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1353   ∈ wcel 2148  {cab 2163  ∀wral 2455  ∃wrex 2456  Vcvv 2737   ↦ cmpt 4064  ran crn 4627   Fn wfn 5211  –onto→wfo 5214  ‘cfv 5216  (class class class)co 5874  [cec 6532   / cqs 6533  Basecbs 12461   /_s cqus 12720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-fun 5218  df-fn 5219  df-fo 5222  df-ec 6536  df-qs 6540

This theorem is referenced by:  qusbas  12746  qusaddvallemg  12751  qusaddflemg  12752  qusaddval  12753  qusaddf  12754  qusmulval  12755  qusmulf  12756