Theorem fo2ndf 6206

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 5347	. . . 4 |- ( F : A --> B -> F Fn A )
2		dffn3 5358	. . . 4 |- ( F Fn A <-> F : A --> ran F )
3	1, 2	sylib 121	. . 3 |- ( F : A --> B -> F : A --> ran F )
4		f2ndf 6205	. . 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 5356	. . . 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 4801	. . . . . 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 5520	. . . . . . . . . 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 2733	. . . . . . . . . 10 |- x e. _V
15		vex 2733	. . . . . . . . . 10 |- y e. _V
16	14, 15	op2nd 6126	. . . . . . . . 9 |- ( 2nd ` <. x , y >. ) = y
17	13, 16	eqtr2di 2220	. . . . . . . 8 |- ( ( F : A --> B /\ <. x , y >. e. F ) -> y = ( ( 2nd |` F ) ` <. x , y >. ) )
18		f2ndf 6205	. . . . . . . . . 10 |- ( F : A --> B -> ( 2nd |` F ) : F --> B )
19		ffn 5347	. . . . . . . . . 10 |- ( ( 2nd |` F ) : F --> B -> ( 2nd |` F ) Fn F )
20	18, 19	syl 14	. . . . . . . . 9 |- ( F : A --> B -> ( 2nd |` F ) Fn F )
21		fnfvelrn 5628	. . . . . . . . 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 2247	. . . . . . 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 1812	. . . . 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 3153	. . 3 |- ( F : A --> B -> ran F C_ ran ( 2nd |` F ) )
28	9, 27	eqssd 3164	. 2 |- ( F : A --> B -> ran ( 2nd |` F ) = ran F )
29		dffo2 5424	. 2 |- ( ( 2nd |` F ) : F -onto-> ran F <-> ( ( 2nd |` F ) : F --> ran F /\ ran ( 2nd |` F ) = ran F ) )
30	5, 28, 29	sylanbrc 415	1 |- ( F : A --> B -> ( 2nd |` F ) : F -onto-> ran F )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   E.wex 1485   e. wcel 2141   C_ wss 3121   <.cop 3586   ran crn 4612   |` cres 4613   Fn wfn 5193   -->wf 5194   -onto->wfo 5196   ` cfv 5198   2ndc2nd 6118

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204  df-fv 5206  df-2nd 6120

This theorem is referenced by:  f1o2ndf1  6207