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Theorem f2ndf 5983
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 5923 . . 3  |-  ( 2nd  |`  ( A  X.  B
) ) : ( A  X.  B ) --> B
2 fssxp 5172 . . 3  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
3 fssres 5180 . . 3  |-  ( ( ( 2nd  |`  ( A  X.  B ) ) : ( A  X.  B ) --> B  /\  F  C_  ( A  X.  B ) )  -> 
( ( 2nd  |`  ( A  X.  B ) )  |`  F ) : F --> B )
41, 2, 3sylancr 405 . 2  |-  ( F : A --> B  -> 
( ( 2nd  |`  ( A  X.  B ) )  |`  F ) : F --> B )
5 resabs1 4737 . . . . 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 2093 . . 3  |-  ( F : A --> B  -> 
( 2nd  |`  F )  =  ( ( 2nd  |`  ( A  X.  B
) )  |`  F ) )
87feq1d 5143 . 2  |-  ( F : A --> B  -> 
( ( 2nd  |`  F ) : F --> B  <->  ( ( 2nd  |`  ( A  X.  B ) )  |`  F ) : F --> B ) )
94, 8mpbird 165 1  |-  ( F : A --> B  -> 
( 2nd  |`  F ) : F --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    C_ wss 2999    X. cxp 4434    |` cres 4438   -->wf 5006   2ndc2nd 5902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-iun 3730  df-br 3844  df-opab 3898  df-mpt 3899  df-id 4118  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-fv 5018  df-2nd 5904
This theorem is referenced by:  fo2ndf  5984  f1o2ndf1  5985
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