Theorem fo2ndf 6387

Description: The 2nd (second component of an ordered pair) function restricted to a function F is a function from F onto the range of F. (Contributed by Alexander van der Vekens, 4-Feb-2018.)

Assertion

Ref	Expression
fo2ndf	|- ( F : A --> B -> ( 2nd |` F ) : F -onto-> ran F )

Proof of Theorem fo2ndf

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffn 5479	. . . 4 |- ( F : A --> B -> F Fn A )
2		dffn3 5490	. . . 4 |- ( F Fn A <-> F : A --> ran F )
3	1, 2	sylib 122	. . 3 |- ( F : A --> B -> F : A --> ran F )
4		f2ndf 6386	. . 3 |- ( F : A --> ran F -> ( 2nd |` F ) : F --> ran F )
5	3, 4	syl 14	. 2 |- ( F : A --> B -> ( 2nd |` F ) : F --> ran F )
6	2, 4	sylbi 121	. . . . 5 |- ( F Fn A -> ( 2nd |` F ) : F --> ran F )
7	1, 6	syl 14	. . . 4 |- ( F : A --> B -> ( 2nd |` F ) : F --> ran F )
8		frn 5488	. . . 4 |- ( ( 2nd |` F ) : F --> ran F -> ran ( 2nd |` F ) C_ ran F )
9	7, 8	syl 14	. . 3 |- ( F : A --> B -> ran ( 2nd |` F ) C_ ran F )
10		elrn2g 4918	. . . . . 6 |- ( y e. ran F -> ( y e. ran F <-> E. x <. x , y >. e. F ) )
11	10	ibi 176	. . . . 5 |- ( y e. ran F -> E. x <. x , y >. e. F )
12		fvres 5659	. . . . . . . . . 10 |- ( <. x , y >. e. F -> ( ( 2nd |` F ) ` <. x , y >. ) = ( 2nd ` <. x , y >. ) )
13	12	adantl 277	. . . . . . . . 9 |- ( ( F : A --> B /\ <. x , y >. e. F ) -> ( ( 2nd |` F ) ` <. x , y >. ) = ( 2nd ` <. x , y >. ) )
14		vex 2803	. . . . . . . . . 10 |- x e. _V
15		vex 2803	. . . . . . . . . 10 |- y e. _V
16	14, 15	op2nd 6305	. . . . . . . . 9 |- ( 2nd ` <. x , y >. ) = y
17	13, 16	eqtr2di 2279	. . . . . . . 8 |- ( ( F : A --> B /\ <. x , y >. e. F ) -> y = ( ( 2nd |` F ) ` <. x , y >. ) )
18		f2ndf 6386	. . . . . . . . . 10 |- ( F : A --> B -> ( 2nd |` F ) : F --> B )
19		ffn 5479	. . . . . . . . . 10 |- ( ( 2nd |` F ) : F --> B -> ( 2nd |` F ) Fn F )
20	18, 19	syl 14	. . . . . . . . 9 |- ( F : A --> B -> ( 2nd |` F ) Fn F )
21		fnfvelrn 5775	. . . . . . . . 9 |- ( ( ( 2nd |` F ) Fn F /\ <. x , y >. e. F ) -> ( ( 2nd |` F ) ` <. x , y >. ) e. ran ( 2nd |` F ) )
22	20, 21	sylan 283	. . . . . . . 8 |- ( ( F : A --> B /\ <. x , y >. e. F ) -> ( ( 2nd |` F ) ` <. x , y >. ) e. ran ( 2nd |` F ) )
23	17, 22	eqeltrd 2306	. . . . . . 7 |- ( ( F : A --> B /\ <. x , y >. e. F ) -> y e. ran ( 2nd |` F ) )
24	23	ex 115	. . . . . 6 |- ( F : A --> B -> ( <. x , y >. e. F -> y e. ran ( 2nd |` F ) ) )
25	24	exlimdv 1865	. . . . 5 |- ( F : A --> B -> ( E. x <. x , y >. e. F -> y e. ran ( 2nd |` F ) ) )
26	11, 25	syl5 32	. . . 4 |- ( F : A --> B -> ( y e. ran F -> y e. ran ( 2nd |` F ) ) )
27	26	ssrdv 3231	. . 3 |- ( F : A --> B -> ran F C_ ran ( 2nd |` F ) )
28	9, 27	eqssd 3242	. 2 |- ( F : A --> B -> ran ( 2nd |` F ) = ran F )
29		dffo2 5560	. 2 |- ( ( 2nd |` F ) : F -onto-> ran F <-> ( ( 2nd |` F ) : F --> ran F /\ ran ( 2nd |` F ) = ran F ) )
30	5, 28, 29	sylanbrc 417	1 |- ( F : A --> B -> ( 2nd |` F ) : F -onto-> ran F )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   E.wex 1538   e. wcel 2200   C_ wss 3198   <.cop 3670   ran crn 4724   |` cres 4725   Fn wfn 5319   -->wf 5320   -onto->wfo 5322   ` cfv 5324   2ndc2nd 6297

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fo 5330  df-fv 5332  df-2nd 6299

This theorem is referenced by:  f1o2ndf1  6388