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Theorem qliftfuns 6761
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 2432 . . . . 5  |-  F/_ y <. [ x ] R ,  A >.
3 nfcv 2432 . . . . . 6  |-  F/_ x [ y ] R
4 nfcsb1v 3126 . . . . . 6  |-  F/_ x [_ y  /  x ]_ A
53, 4nfop 3828 . . . . 5  |-  F/_ x <. [ y ] R ,  [_ y  /  x ]_ A >.
6 eceq1 6712 . . . . . 6  |-  ( x  =  y  ->  [ x ] R  =  [
y ] R )
7 csbeq1a 3102 . . . . . 6  |-  ( x  =  y  ->  A  =  [_ y  /  x ]_ A )
86, 7opeq12d 3820 . . . . 5  |-  ( x  =  y  ->  <. [ x ] R ,  A >.  = 
<. [ y ] R ,  [_ y  /  x ]_ A >. )
92, 5, 8cbvmpt 4126 . . . 4  |-  ( x  e.  X  |->  <. [ x ] R ,  A >. )  =  ( y  e.  X  |->  <. [ y ] R ,  [_ y  /  x ]_ A >. )
109rneqi 4921 . . 3  |-  ran  (
x  e.  X  |->  <. [ x ] R ,  A >. )  =  ran  ( y  e.  X  |-> 
<. [ y ] R ,  [_ y  /  x ]_ A >. )
111, 10eqtri 2316 . 2  |-  F  =  ran  ( y  e.  X  |->  <. [ y ] R ,  [_ y  /  x ]_ A >. )
12 qlift.2 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  Y )
1312ralrimiva 2639 . . 3  |-  ( ph  ->  A. x  e.  X  A  e.  Y )
144nfel1 2442 . . . 4  |-  F/ x [_ y  /  x ]_ A  e.  Y
157eleq1d 2362 . . . 4  |-  ( x  =  y  ->  ( A  e.  Y  <->  [_ y  /  x ]_ A  e.  Y
) )
1614, 15rspc 2891 . . 3  |-  ( y  e.  X  ->  ( A. x  e.  X  A  e.  Y  ->  [_ y  /  x ]_ A  e.  Y )
)
1713, 16mpan9 455 . 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 3097 . 2  |-  ( y  =  z  ->  [_ y  /  x ]_ A  = 
[_ z  /  x ]_ A )
2111, 17, 18, 19, 20qliftfun 6759 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    <-> wb 176    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801   [_csb 3094   <.cop 3656   class class class wbr 4039    e. cmpt 4093   ran crn 4706   Fun wfun 5265    Er wer 6673   [cec 6674
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-er 6676  df-ec 6678  df-qs 6682
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