ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  qliftfuns Unicode version

Theorem qliftfuns 6221
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 2194 . . . . 5  |-  F/_ y <. [ x ] R ,  A >.
3 nfcv 2194 . . . . . 6  |-  F/_ x [ y ] R
4 nfcsb1v 2910 . . . . . 6  |-  F/_ x [_ y  /  x ]_ A
53, 4nfop 3593 . . . . 5  |-  F/_ x <. [ y ] R ,  [_ y  /  x ]_ A >.
6 eceq1 6172 . . . . . 6  |-  ( x  =  y  ->  [ x ] R  =  [
y ] R )
7 csbeq1a 2888 . . . . . 6  |-  ( x  =  y  ->  A  =  [_ y  /  x ]_ A )
86, 7opeq12d 3585 . . . . 5  |-  ( x  =  y  ->  <. [ x ] R ,  A >.  = 
<. [ y ] R ,  [_ y  /  x ]_ A >. )
92, 5, 8cbvmpt 3879 . . . 4  |-  ( x  e.  X  |->  <. [ x ] R ,  A >. )  =  ( y  e.  X  |->  <. [ y ] R ,  [_ y  /  x ]_ A >. )
109rneqi 4590 . . 3  |-  ran  (
x  e.  X  |->  <. [ x ] R ,  A >. )  =  ran  ( y  e.  X  |-> 
<. [ y ] R ,  [_ y  /  x ]_ A >. )
111, 10eqtri 2076 . 2  |-  F  =  ran  ( y  e.  X  |->  <. [ y ] R ,  [_ y  /  x ]_ A >. )
12 qlift.2 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  Y )
1312ralrimiva 2409 . . 3  |-  ( ph  ->  A. x  e.  X  A  e.  Y )
144nfel1 2204 . . . 4  |-  F/ x [_ y  /  x ]_ A  e.  Y
157eleq1d 2122 . . . 4  |-  ( x  =  y  ->  ( A  e.  Y  <->  [_ y  /  x ]_ A  e.  Y
) )
1614, 15rspc 2667 . . 3  |-  ( y  e.  X  ->  ( A. x  e.  X  A  e.  Y  ->  [_ y  /  x ]_ A  e.  Y )
)
1713, 16mpan9 269 . 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 2883 . 2  |-  ( y  =  z  ->  [_ y  /  x ]_ A  = 
[_ z  /  x ]_ A )
2111, 17, 18, 19, 20qliftfun 6219 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 101    <-> wb 102   A.wal 1257    = wceq 1259    e. wcel 1409   A.wral 2323   _Vcvv 2574   [_csb 2880   <.cop 3406   class class class wbr 3792    |-> cmpt 3846   ran crn 4374   Fun wfun 4924    Er wer 6134   [cec 6135
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fv 4938  df-er 6137  df-ec 6139  df-qs 6143
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator