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Theorem quslem 16810
 Description: The function in qusval 16809 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 8278 . . . . . 6 ( 𝑊 → [𝑥] ∈ V)
31, 2syl 17 . . . . 5 (𝜑 → [𝑥] ∈ V)
43ralrimivw 3150 . . . 4 (𝜑 → ∀𝑥𝑉 [𝑥] ∈ V)
5 qusval.f . . . . 5 𝐹 = (𝑥𝑉 ↦ [𝑥] )
65fnmpt 6460 . . . 4 (∀𝑥𝑉 [𝑥] ∈ V → 𝐹 Fn 𝑉)
74, 6syl 17 . . 3 (𝜑𝐹 Fn 𝑉)
8 dffn4 6571 . . 3 (𝐹 Fn 𝑉𝐹:𝑉onto→ran 𝐹)
97, 8sylib 221 . 2 (𝜑𝐹:𝑉onto→ran 𝐹)
105rnmpt 5791 . . . 4 ran 𝐹 = {𝑦 ∣ ∃𝑥𝑉 𝑦 = [𝑥] }
11 df-qs 8280 . . . 4 (𝑉 / ) = {𝑦 ∣ ∃𝑥𝑉 𝑦 = [𝑥] }
1210, 11eqtr4i 2824 . . 3 ran 𝐹 = (𝑉 / )
13 foeq3 6563 . . 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 1538   ∈ wcel 2111  {cab 2776  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ↦ cmpt 5110  ran crn 5520   Fn wfn 6319  –onto→wfo 6322  ‘cfv 6324  (class class class)co 7135  [cec 8272   / cqs 8273  Basecbs 16477   /s cqus 16772 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-fun 6326  df-fn 6327  df-fo 6330  df-ec 8276  df-qs 8280 This theorem is referenced by:  qusbas  16812  quss  16813  qusaddvallem  16818  qusaddflem  16819  qusaddval  16820  qusaddf  16821  qusmulval  16822  qusmulf  16823  qusgrp2  18212  qusring2  19369  znzrhfo  20243  qustps  22334  qustgpopn  22732  qustgplem  22733  qustgphaus  22735  qusker  30976  qusscaval  30979  quslmod  30981  quslmhm  30982  qusdimsum  31124
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