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Theorem qliftfuns 6479
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 2256 . . . . 5  |-  F/_ y <. [ x ] R ,  A >.
3 nfcv 2256 . . . . . 6  |-  F/_ x [ y ] R
4 nfcsb1v 3003 . . . . . 6  |-  F/_ x [_ y  /  x ]_ A
53, 4nfop 3689 . . . . 5  |-  F/_ x <. [ y ] R ,  [_ y  /  x ]_ A >.
6 eceq1 6430 . . . . . 6  |-  ( x  =  y  ->  [ x ] R  =  [
y ] R )
7 csbeq1a 2981 . . . . . 6  |-  ( x  =  y  ->  A  =  [_ y  /  x ]_ A )
86, 7opeq12d 3681 . . . . 5  |-  ( x  =  y  ->  <. [ x ] R ,  A >.  = 
<. [ y ] R ,  [_ y  /  x ]_ A >. )
92, 5, 8cbvmpt 3991 . . . 4  |-  ( x  e.  X  |->  <. [ x ] R ,  A >. )  =  ( y  e.  X  |->  <. [ y ] R ,  [_ y  /  x ]_ A >. )
109rneqi 4735 . . 3  |-  ran  (
x  e.  X  |->  <. [ x ] R ,  A >. )  =  ran  ( y  e.  X  |-> 
<. [ y ] R ,  [_ y  /  x ]_ A >. )
111, 10eqtri 2136 . 2  |-  F  =  ran  ( y  e.  X  |->  <. [ y ] R ,  [_ y  /  x ]_ A >. )
12 qlift.2 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  Y )
1312ralrimiva 2480 . . 3  |-  ( ph  ->  A. x  e.  X  A  e.  Y )
144nfel1 2267 . . . 4  |-  F/ x [_ y  /  x ]_ A  e.  Y
157eleq1d 2184 . . . 4  |-  ( x  =  y  ->  ( A  e.  Y  <->  [_ y  /  x ]_ A  e.  Y
) )
1614, 15rspc 2755 . . 3  |-  ( y  e.  X  ->  ( A. x  e.  X  A  e.  Y  ->  [_ y  /  x ]_ A  e.  Y )
)
1713, 16mpan9 277 . 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 2976 . 2  |-  ( y  =  z  ->  [_ y  /  x ]_ A  = 
[_ z  /  x ]_ A )
2111, 17, 18, 19, 20qliftfun 6477 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 103    <-> wb 104   A.wal 1312    = wceq 1314    e. wcel 1463   A.wral 2391   _Vcvv 2658   [_csb 2973   <.cop 3498   class class class wbr 3897    |-> cmpt 3957   ran crn 4508   Fun wfun 5085    Er wer 6392   [cec 6393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099  df-er 6395  df-ec 6397  df-qs 6401
This theorem is referenced by: (None)
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