ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  f2ndf Unicode version

Theorem f2ndf 6251
Description: The  2nd (second component of an ordered pair) function restricted to a function  F is a function from  F into the codomain of  F. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
Assertion
Ref Expression
f2ndf  |-  ( F : A --> B  -> 
( 2nd  |`  F ) : F --> B )

Proof of Theorem f2ndf
StepHypRef Expression
1 f2ndres 6185 . . 3  |-  ( 2nd  |`  ( A  X.  B
) ) : ( A  X.  B ) --> B
2 fssxp 5402 . . 3  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
3 fssres 5410 . . 3  |-  ( ( ( 2nd  |`  ( A  X.  B ) ) : ( A  X.  B ) --> B  /\  F  C_  ( A  X.  B ) )  -> 
( ( 2nd  |`  ( A  X.  B ) )  |`  F ) : F --> B )
41, 2, 3sylancr 414 . 2  |-  ( F : A --> B  -> 
( ( 2nd  |`  ( A  X.  B ) )  |`  F ) : F --> B )
5 resabs1 4954 . . . . 5  |-  ( F 
C_  ( A  X.  B )  ->  (
( 2nd  |`  ( A  X.  B ) )  |`  F )  =  ( 2nd  |`  F )
)
62, 5syl 14 . . . 4  |-  ( F : A --> B  -> 
( ( 2nd  |`  ( A  X.  B ) )  |`  F )  =  ( 2nd  |`  F )
)
76eqcomd 2195 . . 3  |-  ( F : A --> B  -> 
( 2nd  |`  F )  =  ( ( 2nd  |`  ( A  X.  B
) )  |`  F ) )
87feq1d 5371 . 2  |-  ( F : A --> B  -> 
( ( 2nd  |`  F ) : F --> B  <->  ( ( 2nd  |`  ( A  X.  B ) )  |`  F ) : F --> B ) )
94, 8mpbird 167 1  |-  ( F : A --> B  -> 
( 2nd  |`  F ) : F --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    C_ wss 3144    X. cxp 4642    |` cres 4646   -->wf 5231   2ndc2nd 6164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-2nd 6166
This theorem is referenced by:  fo2ndf  6252  f1o2ndf1  6253
  Copyright terms: Public domain W3C validator