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Theorem quslem 17588
Description: The function in qusval 17587 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 8749 . . . . . 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 6708 . . . 4 (∀𝑥𝑉 [𝑥] ∈ V → 𝐹 Fn 𝑉)
74, 6syl 17 . . 3 (𝜑𝐹 Fn 𝑉)
8 dffn4 6826 . . 3 (𝐹 Fn 𝑉𝐹:𝑉onto→ran 𝐹)
97, 8sylib 218 . 2 (𝜑𝐹:𝑉onto→ran 𝐹)
105rnmpt 5968 . . . 4 ran 𝐹 = {𝑦 ∣ ∃𝑥𝑉 𝑦 = [𝑥] }
11 df-qs 8751 . . . 4 (𝑉 / ) = {𝑦 ∣ ∃𝑥𝑉 𝑦 = [𝑥] }
1210, 11eqtr4i 2768 . . 3 ran 𝐹 = (𝑉 / )
13 foeq3 6818 . . 3 (ran 𝐹 = (𝑉 / ) → (𝐹:𝑉onto→ran 𝐹𝐹:𝑉onto→(𝑉 / )))
1412, 13ax-mp 5 . 2 (𝐹:𝑉onto→ran 𝐹𝐹:𝑉onto→(𝑉 / ))
159, 14sylib 218 1 (𝜑𝐹:𝑉onto→(𝑉 / ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  Vcvv 3480  cmpt 5225  ran crn 5686   Fn wfn 6556  ontowfo 6559  cfv 6561  (class class class)co 7431  [cec 8743   / cqs 8744  Basecbs 17247   /s cqus 17550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-fo 6567  df-ec 8747  df-qs 8751
This theorem is referenced by:  qusbas  17590  quss  17591  qusaddvallem  17596  qusaddflem  17597  qusaddval  17598  qusaddf  17599  qusmulval  17600  qusmulf  17601  qusgrp2  19076  qusrng  20177  qusring2  20331  znzrhfo  21566  qustps  23730  qustgpopn  24128  qustgplem  24129  qustgphaus  24131  qusker  33377  qusvsval  33380  quslmod  33386  quslmhm  33387  qusdimsum  33679
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