Theorem quslem 12910

Description: The function in qusval 12909 is a surjection onto a quotient set. (Contributed by Mario Carneiro, 23-Feb-2015.)

Hypotheses

Ref	Expression
qusval.u	|- ( ph -> U = ( R /.s .~ ) )
qusval.v	|- ( ph -> V = ( Base ` R ) )
qusval.f	|- F = ( x e. V |-> [ x ] .~ )
qusval.e	|- ( ph -> .~ e. W )
qusval.r	|- ( ph -> R e. Z )

Assertion

Ref	Expression
quslem	|- ( ph -> F : V -onto-> ( V /. .~ ) )

Distinct variable groups:   x, .~   ph, 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 |- ( ph -> .~ e. W )
2		ecexg 6593	. . . . . 6 |- ( .~ e. W -> [ x ] .~ e. _V )
3	1, 2	syl 14	. . . . 5 |- ( ph -> [ x ] .~ e. _V )
4	3	ralrimivw 2568	. . . 4 |- ( ph -> A. x e. V [ x ] .~ e. _V )
5		qusval.f	. . . . 5 |- F = ( x e. V |-> [ x ] .~ )
6	5	fnmpt 5381	. . . 4 |- ( A. x e. V [ x ] .~ e. _V -> F Fn V )
7	4, 6	syl 14	. . 3 |- ( ph -> F Fn V )
8		dffn4 5483	. . 3 |- ( F Fn V <-> F : V -onto-> ran F )
9	7, 8	sylib 122	. 2 |- ( ph -> F : V -onto-> ran F )
10	5	rnmpt 4911	. . . 4 |- ran F = { y | E. x e. V y = [ x ] .~ }
11		df-qs 6595	. . . 4 |- ( V /. .~ ) = { y | E. x e. V y = [ x ] .~ }
12	10, 11	eqtr4i 2217	. . 3 |- ran F = ( V /. .~ )
13		foeq3 5475	. . 3 |- ( ran F = ( V /. .~ ) -> ( F : V -onto-> ran F <-> F : V -onto-> ( V /. .~ ) ) )
14	12, 13	ax-mp 5	. 2 |- ( F : V -onto-> ran F <-> F : V -onto-> ( V /. .~ ) )
15	9, 14	sylib 122	1 |- ( ph -> F : V -onto-> ( V /. .~ ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2164   {cab 2179   A.wral 2472   E.wrex 2473   _Vcvv 2760   |-> cmpt 4091   ran crn 4661   Fn wfn 5250   -onto->wfo 5253   ` cfv 5255  (class class class)co 5919   [cec 6587   /.cqs 6588   Basecbs 12621   /._s cqus 12886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-fun 5257  df-fn 5258  df-fo 5261  df-ec 6591  df-qs 6595

This theorem is referenced by:  qusbas  12913  qusaddvallemg  12919  qusaddflemg  12920  qusaddval  12921  qusaddf  12922  qusmulval  12923  qusmulf  12924  qusgrp2  13186  qusrng  13457  qusring2  13565  znzrhfo  14147