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Theorem fliftfuns 6028
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 2571 . . . . 5  |-  F/_ y <. A ,  B >.
3 nfcsb1v 3275 . . . . . 6  |-  F/_ x [_ y  /  x ]_ A
4 nfcsb1v 3275 . . . . . 6  |-  F/_ x [_ y  /  x ]_ B
53, 4nfop 3992 . . . . 5  |-  F/_ x <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >.
6 csbeq1a 3251 . . . . . 6  |-  ( x  =  y  ->  A  =  [_ y  /  x ]_ A )
7 csbeq1a 3251 . . . . . 6  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
86, 7opeq12d 3984 . . . . 5  |-  ( x  =  y  ->  <. A ,  B >.  =  <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >. )
92, 5, 8cbvmpt 4291 . . . 4  |-  ( x  e.  X  |->  <. A ,  B >. )  =  ( y  e.  X  |->  <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >. )
109rneqi 5088 . . 3  |-  ran  (
x  e.  X  |->  <. A ,  B >. )  =  ran  ( y  e.  X  |->  <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >. )
111, 10eqtri 2455 . 2  |-  F  =  ran  ( y  e.  X  |->  <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >. )
12 flift.2 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  R )
1312ralrimiva 2781 . . 3  |-  ( ph  ->  A. x  e.  X  A  e.  R )
143nfel1 2581 . . . 4  |-  F/ x [_ y  /  x ]_ A  e.  R
156eleq1d 2501 . . . 4  |-  ( x  =  y  ->  ( A  e.  R  <->  [_ y  /  x ]_ A  e.  R
) )
1614, 15rspc 3038 . . 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 2781 . . 3  |-  ( ph  ->  A. x  e.  X  B  e.  S )
204nfel1 2581 . . . 4  |-  F/ x [_ y  /  x ]_ B  e.  S
217eleq1d 2501 . . . 4  |-  ( x  =  y  ->  ( B  e.  S  <->  [_ y  /  x ]_ B  e.  S
) )
2220, 21rspc 3038 . . 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 3246 . 2  |-  ( y  =  z  ->  [_ y  /  x ]_ A  = 
[_ z  /  x ]_ A )
25 csbeq1 3246 . 2  |-  ( y  =  z  ->  [_ y  /  x ]_ B  = 
[_ z  /  x ]_ B )
2611, 17, 23, 24, 25fliftfun 6026 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 1652    e. wcel 1725   A.wral 2697   [_csb 3243   <.cop 3809    e. cmpt 4258   ran crn 4871   Fun wfun 5440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454
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