Theorem f2ndf 6123

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 6058	. . 3 |- ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B
2		fssxp 5290	. . 3 |- ( F : A --> B -> F C_ ( A X. B ) )
3		fssres 5298	. . 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 410	. 2 |- ( F : A --> B -> ( ( 2nd |` ( A X. B ) ) |` F ) : F --> B )
5		resabs1 4848	. . . . 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 2145	. . 3 |- ( F : A --> B -> ( 2nd |` F ) = ( ( 2nd |` ( A X. B ) ) |` F ) )
8	7	feq1d 5259	. 2 |- ( F : A --> B -> ( ( 2nd |` F ) : F --> B <-> ( ( 2nd |` ( A X. B ) ) |` F ) : F --> B ) )
9	4, 8	mpbird 166	1 |- ( F : A --> B -> ( 2nd |` F ) : F --> B )

Syntax hints:   -> wi 4   = wceq 1331   C_ wss 3071   X. cxp 4537   |` cres 4541   -->wf 5119   2ndc2nd 6037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-2nd 6039

This theorem is referenced by:  fo2ndf  6124  f1o2ndf1  6125