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Theorem fliftfuns 5977
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 2525 . . . . 5  |-  F/_ y <. A ,  B >.
3 nfcsb1v 3228 . . . . . 6  |-  F/_ x [_ y  /  x ]_ A
4 nfcsb1v 3228 . . . . . 6  |-  F/_ x [_ y  /  x ]_ B
53, 4nfop 3944 . . . . 5  |-  F/_ x <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >.
6 csbeq1a 3204 . . . . . 6  |-  ( x  =  y  ->  A  =  [_ y  /  x ]_ A )
7 csbeq1a 3204 . . . . . 6  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
86, 7opeq12d 3936 . . . . 5  |-  ( x  =  y  ->  <. A ,  B >.  =  <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >. )
92, 5, 8cbvmpt 4242 . . . 4  |-  ( x  e.  X  |->  <. A ,  B >. )  =  ( y  e.  X  |->  <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >. )
109rneqi 5038 . . 3  |-  ran  (
x  e.  X  |->  <. A ,  B >. )  =  ran  ( y  e.  X  |->  <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >. )
111, 10eqtri 2409 . 2  |-  F  =  ran  ( y  e.  X  |->  <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >. )
12 flift.2 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  R )
1312ralrimiva 2734 . . 3  |-  ( ph  ->  A. x  e.  X  A  e.  R )
143nfel1 2535 . . . 4  |-  F/ x [_ y  /  x ]_ A  e.  R
156eleq1d 2455 . . . 4  |-  ( x  =  y  ->  ( A  e.  R  <->  [_ y  /  x ]_ A  e.  R
) )
1614, 15rspc 2991 . . 3  |-  ( y  e.  X  ->  ( A. x  e.  X  A  e.  R  ->  [_ y  /  x ]_ A  e.  R )
)
1713, 16mpan9 456 . 2  |-  ( (
ph  /\  y  e.  X )  ->  [_ y  /  x ]_ A  e.  R )
18 flift.3 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  S )
1918ralrimiva 2734 . . 3  |-  ( ph  ->  A. x  e.  X  B  e.  S )
204nfel1 2535 . . . 4  |-  F/ x [_ y  /  x ]_ B  e.  S
217eleq1d 2455 . . . 4  |-  ( x  =  y  ->  ( B  e.  S  <->  [_ y  /  x ]_ B  e.  S
) )
2220, 21rspc 2991 . . 3  |-  ( y  e.  X  ->  ( A. x  e.  X  B  e.  S  ->  [_ y  /  x ]_ B  e.  S )
)
2319, 22mpan9 456 . 2  |-  ( (
ph  /\  y  e.  X )  ->  [_ y  /  x ]_ B  e.  S )
24 csbeq1 3199 . 2  |-  ( y  =  z  ->  [_ y  /  x ]_ A  = 
[_ z  /  x ]_ A )
25 csbeq1 3199 . 2  |-  ( y  =  z  ->  [_ y  /  x ]_ B  = 
[_ z  /  x ]_ B )
2611, 17, 23, 24, 25fliftfun 5975 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651   [_csb 3196   <.cop 3762    e. cmpt 4209   ran crn 4821   Fun wfun 5390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404
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