Theorem fo2ndf 7284

Description: The 2nd (second member of an ordered pair) function restricted to a function F is a function of 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 6045	. . . 4 |- ( F : A --> B -> F Fn A )
2		dffn3 6054	. . . 4 |- ( F Fn A <-> F : A --> ran F )
3	1, 2	sylib 208	. . 3 |- ( F : A --> B -> F : A --> ran F )
4		f2ndf 7283	. . 3 |- ( F : A --> ran F -> ( 2nd |` F ) : F --> ran F )
5	3, 4	syl 17	. 2 |- ( F : A --> B -> ( 2nd |` F ) : F --> ran F )
6	2, 4	sylbi 207	. . . . 5 |- ( F Fn A -> ( 2nd |` F ) : F --> ran F )
7	1, 6	syl 17	. . . 4 |- ( F : A --> B -> ( 2nd |` F ) : F --> ran F )
8		frn 6053	. . . 4 |- ( ( 2nd |` F ) : F --> ran F -> ran ( 2nd |` F ) C_ ran F )
9	7, 8	syl 17	. . 3 |- ( F : A --> B -> ran ( 2nd |` F ) C_ ran F )
10		elrn2g 5313	. . . . . 6 |- ( y e. ran F -> ( y e. ran F <-> E. x <. x , y >. e. F ) )
11	10	ibi 256	. . . . 5 |- ( y e. ran F -> E. x <. x , y >. e. F )
12		fvres 6207	. . . . . . . . . 10 |- ( <. x , y >. e. F -> ( ( 2nd |` F ) ` <. x , y >. ) = ( 2nd ` <. x , y >. ) )
13	12	adantl 482	. . . . . . . . 9 |- ( ( F : A --> B /\ <. x , y >. e. F ) -> ( ( 2nd |` F ) ` <. x , y >. ) = ( 2nd ` <. x , y >. ) )
14		vex 3203	. . . . . . . . . 10 |- x e. _V
15		vex 3203	. . . . . . . . . 10 |- y e. _V
16	14, 15	op2nd 7177	. . . . . . . . 9 |- ( 2nd ` <. x , y >. ) = y
17	13, 16	syl6req 2673	. . . . . . . 8 |- ( ( F : A --> B /\ <. x , y >. e. F ) -> y = ( ( 2nd |` F ) ` <. x , y >. ) )
18		f2ndf 7283	. . . . . . . . . 10 |- ( F : A --> B -> ( 2nd |` F ) : F --> B )
19		ffn 6045	. . . . . . . . . 10 |- ( ( 2nd |` F ) : F --> B -> ( 2nd |` F ) Fn F )
20	18, 19	syl 17	. . . . . . . . 9 |- ( F : A --> B -> ( 2nd |` F ) Fn F )
21		fnfvelrn 6356	. . . . . . . . 9 |- ( ( ( 2nd |` F ) Fn F /\ <. x , y >. e. F ) -> ( ( 2nd |` F ) ` <. x , y >. ) e. ran ( 2nd |` F ) )
22	20, 21	sylan 488	. . . . . . . 8 |- ( ( F : A --> B /\ <. x , y >. e. F ) -> ( ( 2nd |` F ) ` <. x , y >. ) e. ran ( 2nd |` F ) )
23	17, 22	eqeltrd 2701	. . . . . . 7 |- ( ( F : A --> B /\ <. x , y >. e. F ) -> y e. ran ( 2nd |` F ) )
24	23	ex 450	. . . . . 6 |- ( F : A --> B -> ( <. x , y >. e. F -> y e. ran ( 2nd |` F ) ) )
25	24	exlimdv 1861	. . . . 5 |- ( F : A --> B -> ( E. x <. x , y >. e. F -> y e. ran ( 2nd |` F ) ) )
26	11, 25	syl5 34	. . . 4 |- ( F : A --> B -> ( y e. ran F -> y e. ran ( 2nd |` F ) ) )
27	26	ssrdv 3609	. . 3 |- ( F : A --> B -> ran F C_ ran ( 2nd |` F ) )
28	9, 27	eqssd 3620	. 2 |- ( F : A --> B -> ran ( 2nd |` F ) = ran F )
29		dffo2 6119	. 2 |- ( ( 2nd |` F ) : F -onto-> ran F <-> ( ( 2nd |` F ) : F --> ran F /\ ran ( 2nd |` F ) = ran F ) )
30	5, 28, 29	sylanbrc 698	1 |- ( F : A --> B -> ( 2nd |` F ) : F -onto-> ran F )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   C_ wss 3574   <.cop 4183   ran crn 5115   |` cres 5116   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888   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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-fo 5894  df-fv 5896  df-2nd 7169

This theorem is referenced by:  f1o2ndf1  7285