Theorem f2ndf 6400

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

Step	Hyp	Ref	Expression
1		f2ndres 6332	. . 3 |- ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B
2		fssxp 5510	. . 3 |- ( F : A --> B -> F C_ ( A X. B ) )
3		fssres 5520	. . 3 |- ( ( ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B /\ F C_ ( A X. B ) ) -> ( ( 2nd |` ( A X. B ) ) |` F ) : F --> B )
4	1, 2, 3	sylancr 414	. 2 |- ( F : A --> B -> ( ( 2nd |` ( A X. B ) ) |` F ) : F --> B )
5		resabs1 5048	. . . . 5 |- ( F C_ ( A X. B ) -> ( ( 2nd |` ( A X. B ) ) |` F ) = ( 2nd |` F ) )
6	2, 5	syl 14	. . . 4 |- ( F : A --> B -> ( ( 2nd |` ( A X. B ) ) |` F ) = ( 2nd |` F ) )
7	6	eqcomd 2237	. . 3 |- ( F : A --> B -> ( 2nd |` F ) = ( ( 2nd |` ( A X. B ) ) |` F ) )
8	7	feq1d 5476	. 2 |- ( F : A --> B -> ( ( 2nd |` F ) : F --> B <-> ( ( 2nd |` ( A X. B ) ) |` F ) : F --> B ) )
9	4, 8	mpbird 167	1 |- ( F : A --> B -> ( 2nd |` F ) : F --> B )

Syntax hints:   -> wi 4   = wceq 1398   C_ wss 3201   X. cxp 4729   |` cres 4733   -->wf 5329   2ndc2nd 6311

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-2nd 6313

This theorem is referenced by:  fo2ndf  6401  f1o2ndf1  6402