Theorem quslem 17765

Description: The function in qusval 17764 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 8759	. . . . . 6 ⊢ ( ∼ ∈ 𝑊 → [𝑥] ∼ ∈ V)
3	1, 2	syl 18	. . . . 5 ⊢ (𝜑 → [𝑥] ∼ ∈ V)
4	3	ralrimivw 3162	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V)
5		qusval.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
6	5	fnmpt 6709	. . . 4 ⊢ (∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V → 𝐹 Fn 𝑉)
7	4, 6	syl 18	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝑉)
8		dffn4 6832	. . 3 ⊢ (𝐹 Fn 𝑉 ↔ 𝐹:𝑉⟶ontoran 𝐹)
9	7, 8	sylib 221	. 2 ⊢ (𝜑 → 𝐹:𝑉⟶ontoran 𝐹)
10	5	rnmpt 5951	. . . 4 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝑉 𝑦 = [𝑥] ∼ }
11		df-qs 8761	. . . 4 ⊢ (𝑉 / ∼ ) = {𝑦 ∣ ∃𝑥 ∈ 𝑉 𝑦 = [𝑥] ∼ }
12	10, 11	eqtr4i 2789	. . 3 ⊢ ran 𝐹 = (𝑉 / ∼ )
13		foeq3 6824	. . 3 ⊢ (ran 𝐹 = (𝑉 / ∼ ) → (𝐹:𝑉⟶ontoran 𝐹 ↔ 𝐹:𝑉⟶onto(𝑉 / ∼ )))
14	12, 13	ax-mp 5	. 2 ⊢ (𝐹:𝑉⟶ontoran 𝐹 ↔ 𝐹:𝑉⟶onto(𝑉 / ∼ ))
15	9, 14	sylib 221	1 ⊢ (𝜑 → 𝐹:𝑉⟶onto(𝑉 / ∼ ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ↦ cmpt 5192  ran crn 5659   Fn wfn 6561  ⟶ontowfo 6564  ‘cfv 6566  (class class class)co 7451  [cec 8753   / cqs 8754  Basecbs 17431   /_s cqus 17727

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-fun 6568  df-fn 6569  df-fo 6572  df-ec 8757  df-qs 8761

This theorem is used by:  qusbas  17767  quss  17768  qusaddvallem  17773  qusaddflem  17774  qusaddval  17775  qusaddf  17776  qusmulval  17777  qusmulf  17778  qusmgm  18914  qusmnd  19025  qusgrp2  19318  qusrng  20453  qusring2  20615  znzrhfo  21908  qustps  24096  qustgpopn  24494  qustgplem  24495  qustgphaus  24497  qusker  34014  qusvsval  34017  quslmod  34023  quslmhm  34024  qusdimsum  34371