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Theorem quslem 17048
Description: The function in qusval 17047 is a surjection onto a quotient set. (Contributed by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
qusval.u (𝜑𝑈 = (𝑅 /s ))
qusval.v (𝜑𝑉 = (Base‘𝑅))
qusval.f 𝐹 = (𝑥𝑉 ↦ [𝑥] )
qusval.e (𝜑𝑊)
qusval.r (𝜑𝑅𝑍)
Assertion
Ref Expression
quslem (𝜑𝐹:𝑉onto→(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.
StepHypRef Expression
1 qusval.e . . . . . 6 (𝜑𝑊)
2 ecexg 8395 . . . . . 6 ( 𝑊 → [𝑥] ∈ V)
31, 2syl 17 . . . . 5 (𝜑 → [𝑥] ∈ V)
43ralrimivw 3106 . . . 4 (𝜑 → ∀𝑥𝑉 [𝑥] ∈ V)
5 qusval.f . . . . 5 𝐹 = (𝑥𝑉 ↦ [𝑥] )
65fnmpt 6518 . . . 4 (∀𝑥𝑉 [𝑥] ∈ V → 𝐹 Fn 𝑉)
74, 6syl 17 . . 3 (𝜑𝐹 Fn 𝑉)
8 dffn4 6639 . . 3 (𝐹 Fn 𝑉𝐹:𝑉onto→ran 𝐹)
97, 8sylib 221 . 2 (𝜑𝐹:𝑉onto→ran 𝐹)
105rnmpt 5824 . . . 4 ran 𝐹 = {𝑦 ∣ ∃𝑥𝑉 𝑦 = [𝑥] }
11 df-qs 8397 . . . 4 (𝑉 / ) = {𝑦 ∣ ∃𝑥𝑉 𝑦 = [𝑥] }
1210, 11eqtr4i 2768 . . 3 ran 𝐹 = (𝑉 / )
13 foeq3 6631 . . 3 (ran 𝐹 = (𝑉 / ) → (𝐹:𝑉onto→ran 𝐹𝐹:𝑉onto→(𝑉 / )))
1412, 13ax-mp 5 . 2 (𝐹:𝑉onto→ran 𝐹𝐹:𝑉onto→(𝑉 / ))
159, 14sylib 221 1 (𝜑𝐹:𝑉onto→(𝑉 / ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1543  wcel 2110  {cab 2714  wral 3061  wrex 3062  Vcvv 3408  cmpt 5135  ran crn 5552   Fn wfn 6375  ontowfo 6378  cfv 6380  (class class class)co 7213  [cec 8389   / cqs 8390  Basecbs 16760   /s cqus 17010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-fun 6382  df-fn 6383  df-fo 6386  df-ec 8393  df-qs 8397
This theorem is referenced by:  qusbas  17050  quss  17051  qusaddvallem  17056  qusaddflem  17057  qusaddval  17058  qusaddf  17059  qusmulval  17060  qusmulf  17061  qusgrp2  18481  qusring2  19638  znzrhfo  20512  qustps  22619  qustgpopn  23017  qustgplem  23018  qustgphaus  23020  qusker  31263  qusscaval  31266  quslmod  31268  quslmhm  31269  qusdimsum  31423
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