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Theorem quslem 16472
Description: The function in qusval 16471 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 7953 . . . . . 6 ( 𝑊 → [𝑥] ∈ V)
31, 2syl 17 . . . . 5 (𝜑 → [𝑥] ∈ V)
43ralrimivw 3114 . . . 4 (𝜑 → ∀𝑥𝑉 [𝑥] ∈ V)
5 qusval.f . . . . 5 𝐹 = (𝑥𝑉 ↦ [𝑥] )
65fnmpt 6200 . . . 4 (∀𝑥𝑉 [𝑥] ∈ V → 𝐹 Fn 𝑉)
74, 6syl 17 . . 3 (𝜑𝐹 Fn 𝑉)
8 dffn4 6306 . . 3 (𝐹 Fn 𝑉𝐹:𝑉onto→ran 𝐹)
97, 8sylib 209 . 2 (𝜑𝐹:𝑉onto→ran 𝐹)
105rnmpt 5542 . . . 4 ran 𝐹 = {𝑦 ∣ ∃𝑥𝑉 𝑦 = [𝑥] }
11 df-qs 7955 . . . 4 (𝑉 / ) = {𝑦 ∣ ∃𝑥𝑉 𝑦 = [𝑥] }
1210, 11eqtr4i 2790 . . 3 ran 𝐹 = (𝑉 / )
13 foeq3 6298 . . 3 (ran 𝐹 = (𝑉 / ) → (𝐹:𝑉onto→ran 𝐹𝐹:𝑉onto→(𝑉 / )))
1412, 13ax-mp 5 . 2 (𝐹:𝑉onto→ran 𝐹𝐹:𝑉onto→(𝑉 / ))
159, 14sylib 209 1 (𝜑𝐹:𝑉onto→(𝑉 / ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197   = wceq 1652  wcel 2155  {cab 2751  wral 3055  wrex 3056  Vcvv 3350  cmpt 4890  ran crn 5280   Fn wfn 6065  ontowfo 6068  cfv 6070  (class class class)co 6844  [cec 7947   / cqs 7948  Basecbs 16133   /s cqus 16434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064  ax-un 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-fun 6072  df-fn 6073  df-fo 6076  df-ec 7951  df-qs 7955
This theorem is referenced by:  qusbas  16474  quss  16475  qusaddvallem  16480  qusaddflem  16481  qusaddval  16482  qusaddf  16483  qusmulval  16484  qusmulf  16485  qusgrp2  17803  qusring2  18890  znzrhfo  20171  qustps  21808  qustgpopn  22205  qustgplem  22206  qustgphaus  22208
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