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Theorem qliftfuns 6621
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 )
Assertion
Ref Expression
qliftfuns  |-  ( ph  ->  ( Fun  F  <->  A. y A. z ( y R z  ->  [_ y  /  x ]_ A  =  [_ z  /  x ]_ A
) ) )
Distinct variable groups:    y, z, A   
x, y, z, ph    x, R, y, z    y, F, z    x, X, y, z    x, Y, y, z
Allowed substitution hints:    A( x)    F( x)

Proof of Theorem qliftfuns
StepHypRef Expression
1 qlift.1 . . 3  |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )
2 nfcv 2319 . . . . 5  |-  F/_ y <. [ x ] R ,  A >.
3 nfcv 2319 . . . . . 6  |-  F/_ x [ y ] R
4 nfcsb1v 3092 . . . . . 6  |-  F/_ x [_ y  /  x ]_ A
53, 4nfop 3796 . . . . 5  |-  F/_ x <. [ y ] R ,  [_ y  /  x ]_ A >.
6 eceq1 6572 . . . . . 6  |-  ( x  =  y  ->  [ x ] R  =  [
y ] R )
7 csbeq1a 3068 . . . . . 6  |-  ( x  =  y  ->  A  =  [_ y  /  x ]_ A )
86, 7opeq12d 3788 . . . . 5  |-  ( x  =  y  ->  <. [ x ] R ,  A >.  = 
<. [ y ] R ,  [_ y  /  x ]_ A >. )
92, 5, 8cbvmpt 4100 . . . 4  |-  ( x  e.  X  |->  <. [ x ] R ,  A >. )  =  ( y  e.  X  |->  <. [ y ] R ,  [_ y  /  x ]_ A >. )
109rneqi 4857 . . 3  |-  ran  (
x  e.  X  |->  <. [ x ] R ,  A >. )  =  ran  ( y  e.  X  |-> 
<. [ y ] R ,  [_ y  /  x ]_ A >. )
111, 10eqtri 2198 . 2  |-  F  =  ran  ( y  e.  X  |->  <. [ y ] R ,  [_ y  /  x ]_ A >. )
12 qlift.2 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  Y )
1312ralrimiva 2550 . . 3  |-  ( ph  ->  A. x  e.  X  A  e.  Y )
144nfel1 2330 . . . 4  |-  F/ x [_ y  /  x ]_ A  e.  Y
157eleq1d 2246 . . . 4  |-  ( x  =  y  ->  ( A  e.  Y  <->  [_ y  /  x ]_ A  e.  Y
) )
1614, 15rspc 2837 . . 3  |-  ( y  e.  X  ->  ( A. x  e.  X  A  e.  Y  ->  [_ y  /  x ]_ A  e.  Y )
)
1713, 16mpan9 281 . 2  |-  ( (
ph  /\  y  e.  X )  ->  [_ y  /  x ]_ A  e.  Y )
18 qlift.3 . 2  |-  ( ph  ->  R  Er  X )
19 qlift.4 . 2  |-  ( ph  ->  X  e.  _V )
20 csbeq1 3062 . 2  |-  ( y  =  z  ->  [_ y  /  x ]_ A  = 
[_ z  /  x ]_ A )
2111, 17, 18, 19, 20qliftfun 6619 1  |-  ( ph  ->  ( Fun  F  <->  A. y A. z ( y R z  ->  [_ y  /  x ]_ A  =  [_ z  /  x ]_ A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1351    = wceq 1353    e. wcel 2148   A.wral 2455   _Vcvv 2739   [_csb 3059   <.cop 3597   class class class wbr 4005    |-> cmpt 4066   ran crn 4629   Fun wfun 5212    Er wer 6534   [cec 6535
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-er 6537  df-ec 6539  df-qs 6543
This theorem is referenced by: (None)
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