Theorem qliftf 6788

Description: The domain and codomain 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 6781	. . 3 |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) )
6	1, 5, 2	fliftf 5939	. 2 |- ( ph -> ( Fun F <-> F : ran ( x e. X |-> [ x ] R ) --> Y ) )
7		df-qs 6707	. . . . 5 |- ( X /. R ) = { y | E. x e. X y = [ x ] R }
8		eqid 2231	. . . . . 6 |- ( x e. X |-> [ x ] R ) = ( x e. X |-> [ x ] R )
9	8	rnmpt 4980	. . . . 5 |- ran ( x e. X |-> [ x ] R ) = { y | E. x e. X y = [ x ] R }
10	7, 9	eqtr4i 2255	. . . 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 5470	. 2 |- ( ph -> ( F : ( X /. R ) --> Y <-> F : ran ( x e. X |-> [ x ] R ) --> Y ) )
13	6, 12	bitr4d 191	1 |- ( ph -> ( Fun F <-> F : ( X /. R ) --> Y ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   {cab 2217   E.wrex 2511   _Vcvv 2802   <.cop 3672   |-> cmpt 4150   ran crn 4726   Fun wfun 5320   -->wf 5322   Er wer 6698   [cec 6699   /.cqs 6700

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-er 6701  df-ec 6703  df-qs 6707

This theorem is referenced by: (None)