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Theorem qliftf 7339
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.
StepHypRef 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 )
51, 2, 3, 4qliftlem 7332 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
61, 5, 2fliftf 6136 . 2  |-  ( ph  ->  ( Fun  F  <->  F : ran  ( x  e.  X  |->  [ x ] R
) --> Y ) )
7 df-qs 7257 . . . . 5  |-  ( X /. R )  =  { y  |  E. x  e.  X  y  =  [ x ] R }
8 eqid 2396 . . . . . 6  |-  ( x  e.  X  |->  [ x ] R )  =  ( x  e.  X  |->  [ x ] R )
98rnmpt 5178 . . . . 5  |-  ran  (
x  e.  X  |->  [ x ] R )  =  { y  |  E. x  e.  X  y  =  [ x ] R }
107, 9eqtr4i 2428 . . . 4  |-  ( X /. R )  =  ran  ( x  e.  X  |->  [ x ] R )
1110a1i 11 . . 3  |-  ( ph  ->  ( X /. R
)  =  ran  (
x  e.  X  |->  [ x ] R ) )
1211feq2d 5643 . 2  |-  ( ph  ->  ( F : ( X /. R ) --> Y  <->  F : ran  (
x  e.  X  |->  [ x ] R ) --> Y ) )
136, 12bitr4d 256 1  |-  ( ph  ->  ( Fun  F  <->  F :
( X /. R
) --> Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1836   {cab 2381   E.wrex 2747   _Vcvv 3051   <.cop 3967    |-> cmpt 4442   ran crn 4931   Fun wfun 5507   -->wf 5509    Er wer 7248   [cec 7249   /.cqs 7250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-rab 2755  df-v 3053  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-fv 5521  df-er 7251  df-ec 7253  df-qs 7257
This theorem is referenced by:  orbsta  16491  frgpupf  16931
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