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Theorem qliftfund 7449
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 436 . . 3  |-  ( ph  ->  ( x R y  ->  A  =  B ) )
32alrimivv 1774 . 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 7448 . 2  |-  ( ph  ->  ( Fun  F  <->  A. x A. y ( x R y  ->  A  =  B ) ) )
103, 9mpbird 236 1  |-  ( ph  ->  Fun  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371   A.wal 1442    = wceq 1444    e. wcel 1887   _Vcvv 3045   <.cop 3974   class class class wbr 4402    |-> cmpt 4461   ran crn 4835   Fun wfun 5576    Er wer 7360   [cec 7361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fv 5590  df-er 7363  df-ec 7365  df-qs 7369
This theorem is referenced by:  orbstafun  16965  frgpupf  17423
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