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Theorem qliftfund 6992
Description: The function  F is the unique function defined by  F `  [
x ]  =  A, provided that the well-definedness condition holds. (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 )
qliftfun.4  |-  ( x  =  y  ->  A  =  B )
qliftfund.6  |-  ( (
ph  /\  x R
y )  ->  A  =  B )
Assertion
Ref Expression
qliftfund  |-  ( ph  ->  Fun  F )
Distinct variable groups:    y, A    x, B    x, y, ph    x, R, y    y, F   
x, X, y    x, Y, y
Allowed substitution hints:    A( x)    B( y)    F( x)

Proof of Theorem qliftfund
StepHypRef Expression
1 qliftfund.6 . . . 4  |-  ( (
ph  /\  x R
y )  ->  A  =  B )
21ex 425 . . 3  |-  ( ph  ->  ( x R y  ->  A  =  B ) )
32alrimivv 1643 . 2  |-  ( ph  ->  A. x A. y
( x R y  ->  A  =  B ) )
4 qlift.1 . . 3  |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )
5 qlift.2 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  Y )
6 qlift.3 . . 3  |-  ( ph  ->  R  Er  X )
7 qlift.4 . . 3  |-  ( ph  ->  X  e.  _V )
8 qliftfun.4 . . 3  |-  ( x  =  y  ->  A  =  B )
94, 5, 6, 7, 8qliftfun 6991 . 2  |-  ( ph  ->  ( Fun  F  <->  A. x A. y ( x R y  ->  A  =  B ) ) )
103, 9mpbird 225 1  |-  ( ph  ->  Fun  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   A.wal 1550    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819   class class class wbr 4214    e. cmpt 4268   ran crn 4881   Fun wfun 5450    Er wer 6904   [cec 6905
This theorem is referenced by:  orbstafun  15090  frgpupf  15407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-er 6907  df-ec 6909  df-qs 6913
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