Theorem f2ndf 7283

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

Step	Hyp	Ref	Expression
1		f2ndres 7191	. . 3 |- ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B
2		fssxp 6060	. . 3 |- ( F : A --> B -> F C_ ( A X. B ) )
3		fssres 6070	. . 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 695	. 2 |- ( F : A --> B -> ( ( 2nd |` ( A X. B ) ) |` F ) : F --> B )
5	2	resabs1d 5428	. . . 4 |- ( F : A --> B -> ( ( 2nd |` ( A X. B ) ) |` F ) = ( 2nd |` F ) )
6	5	eqcomd 2628	. . 3 |- ( F : A --> B -> ( 2nd |` F ) = ( ( 2nd |` ( A X. B ) ) |` F ) )
7	6	feq1d 6030	. 2 |- ( F : A --> B -> ( ( 2nd |` F ) : F --> B <-> ( ( 2nd |` ( A X. B ) ) |` F ) : F --> B ) )
8	4, 7	mpbird 247	1 |- ( F : A --> B -> ( 2nd |` F ) : F --> B )

Syntax hints:   -> wi 4   C_ wss 3574   X. cxp 5112   |` cres 5116   -->wf 5884   2ndc2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-2nd 7169

This theorem is referenced by:  fo2ndf  7284  f1o2ndf1  7285