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Theorem qliftfuns 6994
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 2574 . . . . 5  |-  F/_ y <. [ x ] R ,  A >.
3 nfcv 2574 . . . . . 6  |-  F/_ x [ y ] R
4 nfcsb1v 3285 . . . . . 6  |-  F/_ x [_ y  /  x ]_ A
53, 4nfop 4002 . . . . 5  |-  F/_ x <. [ y ] R ,  [_ y  /  x ]_ A >.
6 eceq1 6944 . . . . . 6  |-  ( x  =  y  ->  [ x ] R  =  [
y ] R )
7 csbeq1a 3261 . . . . . 6  |-  ( x  =  y  ->  A  =  [_ y  /  x ]_ A )
86, 7opeq12d 3994 . . . . 5  |-  ( x  =  y  ->  <. [ x ] R ,  A >.  = 
<. [ y ] R ,  [_ y  /  x ]_ A >. )
92, 5, 8cbvmpt 4302 . . . 4  |-  ( x  e.  X  |->  <. [ x ] R ,  A >. )  =  ( y  e.  X  |->  <. [ y ] R ,  [_ y  /  x ]_ A >. )
109rneqi 5099 . . 3  |-  ran  (
x  e.  X  |->  <. [ x ] R ,  A >. )  =  ran  ( y  e.  X  |-> 
<. [ y ] R ,  [_ y  /  x ]_ A >. )
111, 10eqtri 2458 . 2  |-  F  =  ran  ( y  e.  X  |->  <. [ y ] R ,  [_ y  /  x ]_ A >. )
12 qlift.2 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  Y )
1312ralrimiva 2791 . . 3  |-  ( ph  ->  A. x  e.  X  A  e.  Y )
144nfel1 2584 . . . 4  |-  F/ x [_ y  /  x ]_ A  e.  Y
157eleq1d 2504 . . . 4  |-  ( x  =  y  ->  ( A  e.  Y  <->  [_ y  /  x ]_ A  e.  Y
) )
1614, 15rspc 3048 . . 3  |-  ( y  e.  X  ->  ( A. x  e.  X  A  e.  Y  ->  [_ y  /  x ]_ A  e.  Y )
)
1713, 16mpan9 457 . 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 3256 . 2  |-  ( y  =  z  ->  [_ y  /  x ]_ A  = 
[_ z  /  x ]_ A )
2111, 17, 18, 19, 20qliftfun 6992 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 178    /\ wa 360   A.wal 1550    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958   [_csb 3253   <.cop 3819   class class class wbr 4215    e. cmpt 4269   ran crn 4882   Fun wfun 5451    Er wer 6905   [cec 6906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-er 6908  df-ec 6910  df-qs 6914
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