Theorem fo2ndf 6463

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 5533	. . . 4 |- ( F : A --> B -> F Fn A )
2		dffn3 5544	. . . 4 |- ( F Fn A <-> F : A --> ran F )
3	1, 2	sylib 122	. . 3 |- ( F : A --> B -> F : A --> ran F )
4		f2ndf 6462	. . 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 5542	. . . 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 4970	. . . . . 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 5719	. . . . . . . . . 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 2824	. . . . . . . . . 10 |- x e. _V
15		vex 2824	. . . . . . . . . 10 |- y e. _V
16	14, 15	op2nd 6381	. . . . . . . . 9 |- ( 2nd ` <. x , y >. ) = y
17	13, 16	eqtr2di 2288	. . . . . . . 8 |- ( ( F : A --> B /\ <. x , y >. e. F ) -> y = ( ( 2nd |` F ) ` <. x , y >. ) )
18		f2ndf 6462	. . . . . . . . . 10 |- ( F : A --> B -> ( 2nd |` F ) : F --> B )
19		ffn 5533	. . . . . . . . . 10 |- ( ( 2nd |` F ) : F --> B -> ( 2nd |` F ) Fn F )
20	18, 19	syl 14	. . . . . . . . 9 |- ( F : A --> B -> ( 2nd |` F ) Fn F )
21		fnfvelrn 5840	. . . . . . . . 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 2315	. . . . . . 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 1872	. . . . 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 3254	. . 3 |- ( F : A --> B -> ran F C_ ran ( 2nd |` F ) )
28	9, 27	eqssd 3265	. 2 |- ( F : A --> B -> ran ( 2nd |` F ) = ran F )
29		dffo2 5619	. 2 |- ( ( 2nd |` F ) : F -onto-> ran F <-> ( ( 2nd |` F ) : F --> ran F /\ ran ( 2nd |` F ) = ran F ) )
30	5, 28, 29	sylanbrc 421	1 |- ( F : A --> B -> ( 2nd |` F ) : F -onto-> ran F )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   C_ wss 3220   <.cop 3712   ran crn 4775   |` cres 4776   Fn wfn 5372   -->wf 5373   -onto->wfo 5375   ` cfv 5377   2ndc2nd 6373

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fo 5383  df-fv 5385  df-2nd 6375

This theorem is used by:  f1o2ndf1  6464