Theorem f2ndf 6435

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 6367	. . 3 |- ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B
2		fssxp 5535	. . 3 |- ( F : A --> B -> F C_ ( A X. B ) )
3		fssres 5545	. . 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 5072	. . . . 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 2240	. . 3 |- ( F : A --> B -> ( 2nd |` F ) = ( ( 2nd |` ( A X. B ) ) |` F ) )
8	7	feq1d 5500	. 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 3214   X. cxp 4752   |` cres 4756   -->wf 5353   2ndc2nd 6346

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 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-2nd 6348

This theorem is referenced by:  fo2ndf  6436  f1o2ndf1  6437