Theorem qliftf 6562

Description: The domain and range of the function F. (Contributed by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
qlift.1	|- F = ran ( x e. X |-> <. [ x ] R , A >. )
qlift.2	|- ( ( ph /\ x e. X ) -> A e. Y )
qlift.3	|- ( ph -> R Er X )
qlift.4	|- ( ph -> X e. _V )

Assertion

Ref	Expression
qliftf	|- ( ph -> ( Fun F <-> F : ( X /. R ) --> Y ) )

Distinct variable groups:   ph, x   x, R   x, X   x, Y

Allowed substitution hints:   A( x)   F( x)

Proof of Theorem qliftf

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		qlift.1	. . 3 |- F = ran ( x e. X |-> <. [ x ] R , A >. )
2		qlift.2	. . . 4 |- ( ( ph /\ x e. X ) -> A e. Y )
3		qlift.3	. . . 4 |- ( ph -> R Er X )
4		qlift.4	. . . 4 |- ( ph -> X e. _V )
5	1, 2, 3, 4	qliftlem 6555	. . 3 |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) )
6	1, 5, 2	fliftf 5746	. 2 |- ( ph -> ( Fun F <-> F : ran ( x e. X |-> [ x ] R ) --> Y ) )
7		df-qs 6483	. . . . 5 |- ( X /. R ) = { y | E. x e. X y = [ x ] R }
8		eqid 2157	. . . . . 6 |- ( x e. X |-> [ x ] R ) = ( x e. X |-> [ x ] R )
9	8	rnmpt 4833	. . . . 5 |- ran ( x e. X |-> [ x ] R ) = { y | E. x e. X y = [ x ] R }
10	7, 9	eqtr4i 2181	. . . 4 |- ( X /. R ) = ran ( x e. X |-> [ x ] R )
11	10	a1i 9	. . 3 |- ( ph -> ( X /. R ) = ran ( x e. X |-> [ x ] R ) )
12	11	feq2d 5306	. 2 |- ( ph -> ( F : ( X /. R ) --> Y <-> F : ran ( x e. X |-> [ x ] R ) --> Y ) )
13	6, 12	bitr4d 190	1 |- ( ph -> ( Fun F <-> F : ( X /. R ) --> Y ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   {cab 2143   E.wrex 2436   _Vcvv 2712   <.cop 3563   |-> cmpt 4025   ran crn 4586   Fun wfun 5163   -->wf 5165   Er wer 6474   [cec 6475   /.cqs 6476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-fv 5177  df-er 6477  df-ec 6479  df-qs 6483

This theorem is referenced by: (None)