Theorem quslem 12804

Description: The function in qusval 12803 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 6564	. . . . . 6 |- ( .~ e. W -> [ x ] .~ e. _V )
3	1, 2	syl 14	. . . . 5 |- ( ph -> [ x ] .~ e. _V )
4	3	ralrimivw 2564	. . . 4 |- ( ph -> A. x e. V [ x ] .~ e. _V )
5		qusval.f	. . . . 5 |- F = ( x e. V |-> [ x ] .~ )
6	5	fnmpt 5361	. . . 4 |- ( A. x e. V [ x ] .~ e. _V -> F Fn V )
7	4, 6	syl 14	. . 3 |- ( ph -> F Fn V )
8		dffn4 5463	. . 3 |- ( F Fn V <-> F : V -onto-> ran F )
9	7, 8	sylib 122	. 2 |- ( ph -> F : V -onto-> ran F )
10	5	rnmpt 4893	. . . 4 |- ran F = { y | E. x e. V y = [ x ] .~ }
11		df-qs 6566	. . . 4 |- ( V /. .~ ) = { y | E. x e. V y = [ x ] .~ }
12	10, 11	eqtr4i 2213	. . 3 |- ran F = ( V /. .~ )
13		foeq3 5455	. . 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 2160   {cab 2175   A.wral 2468   E.wrex 2469   _Vcvv 2752   |-> cmpt 4079   ran crn 4645   Fn wfn 5230   -onto->wfo 5233   ` cfv 5235  (class class class)co 5897   [cec 6558   /.cqs 6559   Basecbs 12515   /._s cqus 12780

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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-fun 5237  df-fn 5238  df-fo 5241  df-ec 6562  df-qs 6566

This theorem is referenced by:  qusbas  12807  qusaddvallemg  12812  qusaddflemg  12813  qusaddval  12814  qusaddf  12815  qusmulval  12816  qusmulf  12817  qusgrp2  13070  qusrng  13329  qusring2  13433