Theorem quslem 12763

Description: The function in qusval 12762 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 6553	. . . . . 6 ⊢ ( ∼ ∈ 𝑊 → [𝑥] ∼ ∈ V)
3	1, 2	syl 14	. . . . 5 ⊢ (𝜑 → [𝑥] ∼ ∈ V)
4	3	ralrimivw 2561	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V)
5		qusval.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
6	5	fnmpt 5354	. . . 4 ⊢ (∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V → 𝐹 Fn 𝑉)
7	4, 6	syl 14	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝑉)
8		dffn4 5456	. . 3 ⊢ (𝐹 Fn 𝑉 ↔ 𝐹:𝑉–onto→ran 𝐹)
9	7, 8	sylib 122	. 2 ⊢ (𝜑 → 𝐹:𝑉–onto→ran 𝐹)
10	5	rnmpt 4887	. . . 4 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝑉 𝑦 = [𝑥] ∼ }
11		df-qs 6555	. . . 4 ⊢ (𝑉 / ∼ ) = {𝑦 ∣ ∃𝑥 ∈ 𝑉 𝑦 = [𝑥] ∼ }
12	10, 11	eqtr4i 2211	. . 3 ⊢ ran 𝐹 = (𝑉 / ∼ )
13		foeq3 5448	. . 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 1363   ∈ wcel 2158  {cab 2173  ∀wral 2465  ∃wrex 2466  Vcvv 2749   ↦ cmpt 4076  ran crn 4639   Fn wfn 5223  –onto→wfo 5226  ‘cfv 5228  (class class class)co 5888  [cec 6547   / cqs 6548  Basecbs 12476   /_s cqus 12739

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-fun 5230  df-fn 5231  df-fo 5234  df-ec 6551  df-qs 6555

This theorem is referenced by:  qusbas  12766  qusaddvallemg  12771  qusaddflemg  12772  qusaddval  12773  qusaddf  12774  qusmulval  12775  qusmulf  12776  qusgrp2  13008  qusrng  13210  qusring2  13314