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Theorem fliftfuns 5777
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 2312 . . . . 5  |-  F/_ y <. A ,  B >.
3 nfcsb1v 3082 . . . . . 6  |-  F/_ x [_ y  /  x ]_ A
4 nfcsb1v 3082 . . . . . 6  |-  F/_ x [_ y  /  x ]_ B
53, 4nfop 3781 . . . . 5  |-  F/_ x <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >.
6 csbeq1a 3058 . . . . . 6  |-  ( x  =  y  ->  A  =  [_ y  /  x ]_ A )
7 csbeq1a 3058 . . . . . 6  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
86, 7opeq12d 3773 . . . . 5  |-  ( x  =  y  ->  <. A ,  B >.  =  <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >. )
92, 5, 8cbvmpt 4084 . . . 4  |-  ( x  e.  X  |->  <. A ,  B >. )  =  ( y  e.  X  |->  <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >. )
109rneqi 4839 . . 3  |-  ran  (
x  e.  X  |->  <. A ,  B >. )  =  ran  ( y  e.  X  |->  <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >. )
111, 10eqtri 2191 . 2  |-  F  =  ran  ( y  e.  X  |->  <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >. )
12 flift.2 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  R )
1312ralrimiva 2543 . . 3  |-  ( ph  ->  A. x  e.  X  A  e.  R )
143nfel1 2323 . . . 4  |-  F/ x [_ y  /  x ]_ A  e.  R
156eleq1d 2239 . . . 4  |-  ( x  =  y  ->  ( A  e.  R  <->  [_ y  /  x ]_ A  e.  R
) )
1614, 15rspc 2828 . . 3  |-  ( y  e.  X  ->  ( A. x  e.  X  A  e.  R  ->  [_ y  /  x ]_ A  e.  R )
)
1713, 16mpan9 279 . 2  |-  ( (
ph  /\  y  e.  X )  ->  [_ y  /  x ]_ A  e.  R )
18 flift.3 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  S )
1918ralrimiva 2543 . . 3  |-  ( ph  ->  A. x  e.  X  B  e.  S )
204nfel1 2323 . . . 4  |-  F/ x [_ y  /  x ]_ B  e.  S
217eleq1d 2239 . . . 4  |-  ( x  =  y  ->  ( B  e.  S  <->  [_ y  /  x ]_ B  e.  S
) )
2220, 21rspc 2828 . . 3  |-  ( y  e.  X  ->  ( A. x  e.  X  B  e.  S  ->  [_ y  /  x ]_ B  e.  S )
)
2319, 22mpan9 279 . 2  |-  ( (
ph  /\  y  e.  X )  ->  [_ y  /  x ]_ B  e.  S )
24 csbeq1 3052 . 2  |-  ( y  =  z  ->  [_ y  /  x ]_ A  = 
[_ z  /  x ]_ A )
25 csbeq1 3052 . 2  |-  ( y  =  z  ->  [_ y  /  x ]_ B  = 
[_ z  /  x ]_ B )
2611, 17, 23, 24, 25fliftfun 5775 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    /\ wa 103    <-> wb 104    = wceq 1348    e. wcel 2141   A.wral 2448   [_csb 3049   <.cop 3586    |-> cmpt 4050   ran crn 4612   Fun wfun 5192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206
This theorem is referenced by: (None)
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