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Theorem qliftfund 6375
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 113 . . 3  |-  ( ph  ->  ( x R y  ->  A  =  B ) )
32alrimivv 1803 . 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 6374 . 2  |-  ( ph  ->  ( Fun  F  <->  A. x A. y ( x R y  ->  A  =  B ) ) )
103, 9mpbird 165 1  |-  ( ph  ->  Fun  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1287    = wceq 1289    e. wcel 1438   _Vcvv 2619   <.cop 3449   class class class wbr 3845    |-> cmpt 3899   ran crn 4439   Fun wfun 5009    Er wer 6289   [cec 6290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023  df-er 6292  df-ec 6294  df-qs 6298
This theorem is referenced by: (None)
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