Theorem qliftf 6707

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 6700	. . 3 |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) )
6	1, 5, 2	fliftf 5868	. 2 |- ( ph -> ( Fun F <-> F : ran ( x e. X |-> [ x ] R ) --> Y ) )
7		df-qs 6626	. . . . 5 |- ( X /. R ) = { y | E. x e. X y = [ x ] R }
8		eqid 2205	. . . . . 6 |- ( x e. X |-> [ x ] R ) = ( x e. X |-> [ x ] R )
9	8	rnmpt 4926	. . . . 5 |- ran ( x e. X |-> [ x ] R ) = { y | E. x e. X y = [ x ] R }
10	7, 9	eqtr4i 2229	. . . 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 5413	. 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 1373   e. wcel 2176   {cab 2191   E.wrex 2485   _Vcvv 2772   <.cop 3636   |-> cmpt 4105   ran crn 4676   Fun wfun 5265   -->wf 5267   Er wer 6617   [cec 6618   /.cqs 6619

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-er 6620  df-ec 6622  df-qs 6626

This theorem is referenced by: (None)