MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fliftfuns Unicode version

Theorem fliftfuns 5829
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 2432 . . . . 5  |-  F/_ y <. A ,  B >.
3 nfcsb1v 3126 . . . . . 6  |-  F/_ x [_ y  /  x ]_ A
4 nfcsb1v 3126 . . . . . 6  |-  F/_ x [_ y  /  x ]_ B
53, 4nfop 3828 . . . . 5  |-  F/_ x <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >.
6 csbeq1a 3102 . . . . . 6  |-  ( x  =  y  ->  A  =  [_ y  /  x ]_ A )
7 csbeq1a 3102 . . . . . 6  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
86, 7opeq12d 3820 . . . . 5  |-  ( x  =  y  ->  <. A ,  B >.  =  <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >. )
92, 5, 8cbvmpt 4126 . . . 4  |-  ( x  e.  X  |->  <. A ,  B >. )  =  ( y  e.  X  |->  <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >. )
109rneqi 4921 . . 3  |-  ran  (
x  e.  X  |->  <. A ,  B >. )  =  ran  ( y  e.  X  |->  <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >. )
111, 10eqtri 2316 . 2  |-  F  =  ran  ( y  e.  X  |->  <. [_ y  /  x ]_ A ,  [_ y  /  x ]_ B >. )
12 flift.2 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  R )
1312ralrimiva 2639 . . 3  |-  ( ph  ->  A. x  e.  X  A  e.  R )
143nfel1 2442 . . . 4  |-  F/ x [_ y  /  x ]_ A  e.  R
156eleq1d 2362 . . . 4  |-  ( x  =  y  ->  ( A  e.  R  <->  [_ y  /  x ]_ A  e.  R
) )
1614, 15rspc 2891 . . 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 2639 . . 3  |-  ( ph  ->  A. x  e.  X  B  e.  S )
204nfel1 2442 . . . 4  |-  F/ x [_ y  /  x ]_ B  e.  S
217eleq1d 2362 . . . 4  |-  ( x  =  y  ->  ( B  e.  S  <->  [_ y  /  x ]_ B  e.  S
) )
2220, 21rspc 2891 . . 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 3097 . 2  |-  ( y  =  z  ->  [_ y  /  x ]_ A  = 
[_ z  /  x ]_ A )
25 csbeq1 3097 . 2  |-  ( y  =  z  ->  [_ y  /  x ]_ B  = 
[_ z  /  x ]_ B )
2611, 17, 23, 24, 25fliftfun 5827 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 1632    e. wcel 1696   A.wral 2556   [_csb 3094   <.cop 3656    e. cmpt 4093   ran crn 4706   Fun wfun 5265
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279
  Copyright terms: Public domain W3C validator