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Theorem f2ndf 5847
Description: The  2nd (second member of an ordered pair) function restricted to a function  F is a function of  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 5787 . . 3  |-  ( 2nd  |`  ( A  X.  B
) ) : ( A  X.  B ) --> B
2 fssxp 5058 . . 3  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
3 fssres 5066 . . 3  |-  ( ( ( 2nd  |`  ( A  X.  B ) ) : ( A  X.  B ) --> B  /\  F  C_  ( A  X.  B ) )  -> 
( ( 2nd  |`  ( A  X.  B ) )  |`  F ) : F --> B )
41, 2, 3sylancr 393 . 2  |-  ( F : A --> B  -> 
( ( 2nd  |`  ( A  X.  B ) )  |`  F ) : F --> B )
5 resabs1 4640 . . . . 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 2045 . . 3  |-  ( F : A --> B  -> 
( 2nd  |`  F )  =  ( ( 2nd  |`  ( A  X.  B
) )  |`  F ) )
87feq1d 5034 . 2  |-  ( F : A --> B  -> 
( ( 2nd  |`  F ) : F --> B  <->  ( ( 2nd  |`  ( A  X.  B ) )  |`  F ) : F --> B ) )
94, 8mpbird 156 1  |-  ( F : A --> B  -> 
( 2nd  |`  F ) : F --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243    C_ wss 2917    X. cxp 4343    |` cres 4347   -->wf 4898   2ndc2nd 5766
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910  df-2nd 5768
This theorem is referenced by:  fo2ndf  5848  f1o2ndf1  5849
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