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

Step	Hyp	Ref	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 )
4	1, 2, 3	sylancr 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 ) )
6	2, 5	syl 14	. . . 4 |- ( F : A --> B -> ( ( 2nd |` ( A X. B ) ) |` F ) = ( 2nd |` F ) )
7	6	eqcomd 2195	. . 3 |- ( F : A --> B -> ( 2nd |` F ) = ( ( 2nd |` ( A X. B ) ) |` F ) )
8	7	feq1d 5371	. 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 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