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Theorem fliftfuns 5813
Description: The function  F is the unique function defined by  F `  A  =  B, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
flift.1  |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )
flift.2  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  R )
flift.3  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  S )
Assertion
Ref Expression
fliftfuns  |-  ( ph  ->  ( Fun  F  <->  A. y  e.  X  A. z  e.  X  ( [_ y  /  x ]_ A  =  [_ z  /  x ]_ A  ->  [_ y  /  x ]_ B  = 
[_ z  /  x ]_ B ) ) )
Distinct variable groups:    y, z, A    y, B, z    x, z, y, R    y, F, z    ph, x, y, z   
x, X, y, z   
x, S, y, z
Allowed substitution hints:    A( x)    B( x)    F( x)

Proof of Theorem fliftfuns
StepHypRef Expression
1 flift.1 . . 3  |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )
2 nfcv 2419 . . . . 5  |-  F/_ y <. A ,  B >.
3 nfcsb1v 3113 . . . . . 6  |-  F/_ x [_ y  /  x ]_ A
4 nfcsb1v 3113 . . . . . 6  |-  F/_ x [_ y  /  x ]_ B
53, 4nfop 3812 . . . . 5  |-  F/_ x <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >.
6 csbeq1a 3089 . . . . . 6  |-  ( x  =  y  ->  A  =  [_ y  /  x ]_ A )
7 csbeq1a 3089 . . . . . 6  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
86, 7opeq12d 3804 . . . . 5  |-  ( x  =  y  ->  <. A ,  B >.  =  <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >. )
92, 5, 8cbvmpt 4110 . . . 4  |-  ( x  e.  X  |->  <. A ,  B >. )  =  ( y  e.  X  |->  <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >. )
109rneqi 4905 . . 3  |-  ran  (
x  e.  X  |->  <. A ,  B >. )  =  ran  ( y  e.  X  |->  <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >. )
111, 10eqtri 2303 . 2  |-  F  =  ran  ( y  e.  X  |->  <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >. )
12 flift.2 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  R )
1312ralrimiva 2626 . . 3  |-  ( ph  ->  A. x  e.  X  A  e.  R )
143nfel1 2429 . . . 4  |-  F/ x [_ y  /  x ]_ A  e.  R
156eleq1d 2349 . . . 4  |-  ( x  =  y  ->  ( A  e.  R  <->  [_ y  /  x ]_ A  e.  R
) )
1614, 15rspc 2878 . . 3  |-  ( y  e.  X  ->  ( A. x  e.  X  A  e.  R  ->  [_ y  /  x ]_ A  e.  R )
)
1713, 16mpan9 455 . 2  |-  ( (
ph  /\  y  e.  X )  ->  [_ y  /  x ]_ A  e.  R )
18 flift.3 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  S )
1918ralrimiva 2626 . . 3  |-  ( ph  ->  A. x  e.  X  B  e.  S )
204nfel1 2429 . . . 4  |-  F/ x [_ y  /  x ]_ B  e.  S
217eleq1d 2349 . . . 4  |-  ( x  =  y  ->  ( B  e.  S  <->  [_ y  /  x ]_ B  e.  S
) )
2220, 21rspc 2878 . . 3  |-  ( y  e.  X  ->  ( A. x  e.  X  B  e.  S  ->  [_ y  /  x ]_ B  e.  S )
)
2319, 22mpan9 455 . 2  |-  ( (
ph  /\  y  e.  X )  ->  [_ y  /  x ]_ B  e.  S )
24 csbeq1 3084 . 2  |-  ( y  =  z  ->  [_ y  /  x ]_ A  = 
[_ z  /  x ]_ A )
25 csbeq1 3084 . 2  |-  ( y  =  z  ->  [_ y  /  x ]_ B  = 
[_ z  /  x ]_ B )
2611, 17, 23, 24, 25fliftfun 5811 1  |-  ( ph  ->  ( Fun  F  <->  A. y  e.  X  A. z  e.  X  ( [_ y  /  x ]_ A  =  [_ z  /  x ]_ A  ->  [_ y  /  x ]_ B  = 
[_ z  /  x ]_ B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   [_csb 3081   <.cop 3643    e. cmpt 4077   ran crn 4690   Fun wfun 5249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263
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