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Theorem fo2ndf 5992
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.
StepHypRef Expression
1 ffn 5161 . . . 4  |-  ( F : A --> B  ->  F  Fn  A )
2 dffn3 5171 . . . 4  |-  ( F  Fn  A  <->  F : A
--> ran  F )
31, 2sylib 120 . . 3  |-  ( F : A --> B  ->  F : A --> ran  F
)
4 f2ndf 5991 . . 3  |-  ( F : A --> ran  F  ->  ( 2nd  |`  F ) : F --> ran  F
)
53, 4syl 14 . 2  |-  ( F : A --> B  -> 
( 2nd  |`  F ) : F --> ran  F
)
62, 4sylbi 119 . . . . 5  |-  ( F  Fn  A  ->  ( 2nd  |`  F ) : F --> ran  F )
71, 6syl 14 . . . 4  |-  ( F : A --> B  -> 
( 2nd  |`  F ) : F --> ran  F
)
8 frn 5169 . . . 4  |-  ( ( 2nd  |`  F ) : F --> ran  F  ->  ran  ( 2nd  |`  F ) 
C_  ran  F )
97, 8syl 14 . . 3  |-  ( F : A --> B  ->  ran  ( 2nd  |`  F ) 
C_  ran  F )
10 elrn2g 4626 . . . . . 6  |-  ( y  e.  ran  F  -> 
( y  e.  ran  F  <->  E. x <. x ,  y
>.  e.  F ) )
1110ibi 174 . . . . 5  |-  ( y  e.  ran  F  ->  E. x <. x ,  y
>.  e.  F )
12 fvres 5329 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  F  ->  ( ( 2nd  |`  F ) `  <. x ,  y
>. )  =  ( 2nd `  <. x ,  y
>. ) )
1312adantl 271 . . . . . . . . 9  |-  ( ( F : A --> B  /\  <.
x ,  y >.  e.  F )  ->  (
( 2nd  |`  F ) `
 <. x ,  y
>. )  =  ( 2nd `  <. x ,  y
>. ) )
14 vex 2622 . . . . . . . . . 10  |-  x  e. 
_V
15 vex 2622 . . . . . . . . . 10  |-  y  e. 
_V
1614, 15op2nd 5918 . . . . . . . . 9  |-  ( 2nd `  <. x ,  y
>. )  =  y
1713, 16syl6req 2137 . . . . . . . 8  |-  ( ( F : A --> B  /\  <.
x ,  y >.  e.  F )  ->  y  =  ( ( 2nd  |`  F ) `  <. x ,  y >. )
)
18 f2ndf 5991 . . . . . . . . . 10  |-  ( F : A --> B  -> 
( 2nd  |`  F ) : F --> B )
19 ffn 5161 . . . . . . . . . 10  |-  ( ( 2nd  |`  F ) : F --> B  ->  ( 2nd  |`  F )  Fn  F )
2018, 19syl 14 . . . . . . . . 9  |-  ( F : A --> B  -> 
( 2nd  |`  F )  Fn  F )
21 fnfvelrn 5431 . . . . . . . . 9  |-  ( ( ( 2nd  |`  F )  Fn  F  /\  <. x ,  y >.  e.  F
)  ->  ( ( 2nd  |`  F ) `  <. x ,  y >.
)  e.  ran  ( 2nd  |`  F ) )
2220, 21sylan 277 . . . . . . . 8  |-  ( ( F : A --> B  /\  <.
x ,  y >.  e.  F )  ->  (
( 2nd  |`  F ) `
 <. x ,  y
>. )  e.  ran  ( 2nd  |`  F )
)
2317, 22eqeltrd 2164 . . . . . . 7  |-  ( ( F : A --> B  /\  <.
x ,  y >.  e.  F )  ->  y  e.  ran  ( 2nd  |`  F ) )
2423ex 113 . . . . . 6  |-  ( F : A --> B  -> 
( <. x ,  y
>.  e.  F  ->  y  e.  ran  ( 2nd  |`  F ) ) )
2524exlimdv 1747 . . . . 5  |-  ( F : A --> B  -> 
( E. x <. x ,  y >.  e.  F  ->  y  e.  ran  ( 2nd  |`  F ) ) )
2611, 25syl5 32 . . . 4  |-  ( F : A --> B  -> 
( y  e.  ran  F  ->  y  e.  ran  ( 2nd  |`  F )
) )
2726ssrdv 3031 . . 3  |-  ( F : A --> B  ->  ran  F  C_  ran  ( 2nd  |`  F ) )
289, 27eqssd 3042 . 2  |-  ( F : A --> B  ->  ran  ( 2nd  |`  F )  =  ran  F )
29 dffo2 5237 . 2  |-  ( ( 2nd  |`  F ) : F -onto-> ran  F  <->  ( ( 2nd  |`  F ) : F --> ran  F  /\  ran  ( 2nd  |`  F )  =  ran  F ) )
305, 28, 29sylanbrc 408 1  |-  ( F : A --> B  -> 
( 2nd  |`  F ) : F -onto-> ran  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289   E.wex 1426    e. wcel 1438    C_ wss 2999   <.cop 3449   ran crn 4439    |` cres 4440    Fn wfn 5010   -->wf 5011   -onto->wfo 5013   ` cfv 5015   2ndc2nd 5910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fo 5021  df-fv 5023  df-2nd 5912
This theorem is referenced by:  f1o2ndf1  5993
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