Theorem fo2ndf 6195

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 5337	. . . 4 |- ( F : A --> B -> F Fn A )
2		dffn3 5348	. . . 4 |- ( F Fn A <-> F : A --> ran F )
3	1, 2	sylib 121	. . 3 |- ( F : A --> B -> F : A --> ran F )
4		f2ndf 6194	. . 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 120	. . . . 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 5346	. . . 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 4794	. . . . . 6 |- ( y e. ran F -> ( y e. ran F <-> E. x <. x , y >. e. F ) )
11	10	ibi 175	. . . . 5 |- ( y e. ran F -> E. x <. x , y >. e. F )
12		fvres 5510	. . . . . . . . . 10 |- ( <. x , y >. e. F -> ( ( 2nd |` F ) ` <. x , y >. ) = ( 2nd ` <. x , y >. ) )
13	12	adantl 275	. . . . . . . . 9 |- ( ( F : A --> B /\ <. x , y >. e. F ) -> ( ( 2nd |` F ) ` <. x , y >. ) = ( 2nd ` <. x , y >. ) )
14		vex 2729	. . . . . . . . . 10 |- x e. _V
15		vex 2729	. . . . . . . . . 10 |- y e. _V
16	14, 15	op2nd 6115	. . . . . . . . 9 |- ( 2nd ` <. x , y >. ) = y
17	13, 16	eqtr2di 2216	. . . . . . . 8 |- ( ( F : A --> B /\ <. x , y >. e. F ) -> y = ( ( 2nd |` F ) ` <. x , y >. ) )
18		f2ndf 6194	. . . . . . . . . 10 |- ( F : A --> B -> ( 2nd |` F ) : F --> B )
19		ffn 5337	. . . . . . . . . 10 |- ( ( 2nd |` F ) : F --> B -> ( 2nd |` F ) Fn F )
20	18, 19	syl 14	. . . . . . . . 9 |- ( F : A --> B -> ( 2nd |` F ) Fn F )
21		fnfvelrn 5617	. . . . . . . . 9 |- ( ( ( 2nd |` F ) Fn F /\ <. x , y >. e. F ) -> ( ( 2nd |` F ) ` <. x , y >. ) e. ran ( 2nd |` F ) )
22	20, 21	sylan 281	. . . . . . . 8 |- ( ( F : A --> B /\ <. x , y >. e. F ) -> ( ( 2nd |` F ) ` <. x , y >. ) e. ran ( 2nd |` F ) )
23	17, 22	eqeltrd 2243	. . . . . . 7 |- ( ( F : A --> B /\ <. x , y >. e. F ) -> y e. ran ( 2nd |` F ) )
24	23	ex 114	. . . . . 6 |- ( F : A --> B -> ( <. x , y >. e. F -> y e. ran ( 2nd |` F ) ) )
25	24	exlimdv 1807	. . . . 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 3148	. . 3 |- ( F : A --> B -> ran F C_ ran ( 2nd |` F ) )
28	9, 27	eqssd 3159	. 2 |- ( F : A --> B -> ran ( 2nd |` F ) = ran F )
29		dffo2 5414	. 2 |- ( ( 2nd |` F ) : F -onto-> ran F <-> ( ( 2nd |` F ) : F --> ran F /\ ran ( 2nd |` F ) = ran F ) )
30	5, 28, 29	sylanbrc 414	1 |- ( F : A --> B -> ( 2nd |` F ) : F -onto-> ran F )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   E.wex 1480   e. wcel 2136   C_ wss 3116   <.cop 3579   ran crn 4605   |` cres 4606   Fn wfn 5183   -->wf 5184   -onto->wfo 5186   ` cfv 5188   2ndc2nd 6107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fo 5194  df-fv 5196  df-2nd 6109

This theorem is referenced by:  f1o2ndf1  6196