Theorem qliftf 6854

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 6847	. . 3 |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) )
6	1, 5, 2	fliftf 5972	. 2 |- ( ph -> ( Fun F <-> F : ran ( x e. X |-> [ x ] R ) --> Y ) )
7		df-qs 6773	. . . . 5 |- ( X /. R ) = { y | E. x e. X y = [ x ] R }
8		eqid 2232	. . . . . 6 |- ( x e. X |-> [ x ] R ) = ( x e. X |-> [ x ] R )
9	8	rnmpt 5005	. . . . 5 |- ran ( x e. X |-> [ x ] R ) = { y | E. x e. X y = [ x ] R }
10	7, 9	eqtr4i 2256	. . . 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 5496	. 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 1398   e. wcel 2203   {cab 2218   E.wrex 2521   _Vcvv 2813   <.cop 3692   |-> cmpt 4171   ran crn 4750   Fun wfun 5346   -->wf 5348   Er wer 6764   [cec 6765   /.cqs 6766

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-er 6767  df-ec 6769  df-qs 6773

This theorem is referenced by: (None)