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Theorem fo2ndf 6092
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 5242 . . . 4  |-  ( F : A --> B  ->  F  Fn  A )
2 dffn3 5253 . . . 4  |-  ( F  Fn  A  <->  F : A
--> ran  F )
31, 2sylib 121 . . 3  |-  ( F : A --> B  ->  F : A --> ran  F
)
4 f2ndf 6091 . . 3  |-  ( F : A --> ran  F  ->  ( 2nd  |`  F ) : F --> ran  F
)
53, 4syl 14 . 2  |-  ( F : A --> B  -> 
( 2nd  |`  F ) : F --> ran  F
)
62, 4sylbi 120 . . . . 5  |-  ( F  Fn  A  ->  ( 2nd  |`  F ) : F --> ran  F )
71, 6syl 14 . . . 4  |-  ( F : A --> B  -> 
( 2nd  |`  F ) : F --> ran  F
)
8 frn 5251 . . . 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 4699 . . . . . 6  |-  ( y  e.  ran  F  -> 
( y  e.  ran  F  <->  E. x <. x ,  y
>.  e.  F ) )
1110ibi 175 . . . . 5  |-  ( y  e.  ran  F  ->  E. x <. x ,  y
>.  e.  F )
12 fvres 5413 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  F  ->  ( ( 2nd  |`  F ) `  <. x ,  y
>. )  =  ( 2nd `  <. x ,  y
>. ) )
1312adantl 275 . . . . . . . . 9  |-  ( ( F : A --> B  /\  <.
x ,  y >.  e.  F )  ->  (
( 2nd  |`  F ) `
 <. x ,  y
>. )  =  ( 2nd `  <. x ,  y
>. ) )
14 vex 2663 . . . . . . . . . 10  |-  x  e. 
_V
15 vex 2663 . . . . . . . . . 10  |-  y  e. 
_V
1614, 15op2nd 6013 . . . . . . . . 9  |-  ( 2nd `  <. x ,  y
>. )  =  y
1713, 16syl6req 2167 . . . . . . . 8  |-  ( ( F : A --> B  /\  <.
x ,  y >.  e.  F )  ->  y  =  ( ( 2nd  |`  F ) `  <. x ,  y >. )
)
18 f2ndf 6091 . . . . . . . . . 10  |-  ( F : A --> B  -> 
( 2nd  |`  F ) : F --> B )
19 ffn 5242 . . . . . . . . . 10  |-  ( ( 2nd  |`  F ) : F --> B  ->  ( 2nd  |`  F )  Fn  F )
2018, 19syl 14 . . . . . . . . 9  |-  ( F : A --> B  -> 
( 2nd  |`  F )  Fn  F )
21 fnfvelrn 5520 . . . . . . . . 9  |-  ( ( ( 2nd  |`  F )  Fn  F  /\  <. x ,  y >.  e.  F
)  ->  ( ( 2nd  |`  F ) `  <. x ,  y >.
)  e.  ran  ( 2nd  |`  F ) )
2220, 21sylan 281 . . . . . . . 8  |-  ( ( F : A --> B  /\  <.
x ,  y >.  e.  F )  ->  (
( 2nd  |`  F ) `
 <. x ,  y
>. )  e.  ran  ( 2nd  |`  F )
)
2317, 22eqeltrd 2194 . . . . . . 7  |-  ( ( F : A --> B  /\  <.
x ,  y >.  e.  F )  ->  y  e.  ran  ( 2nd  |`  F ) )
2423ex 114 . . . . . 6  |-  ( F : A --> B  -> 
( <. x ,  y
>.  e.  F  ->  y  e.  ran  ( 2nd  |`  F ) ) )
2524exlimdv 1775 . . . . 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 3073 . . 3  |-  ( F : A --> B  ->  ran  F  C_  ran  ( 2nd  |`  F ) )
289, 27eqssd 3084 . 2  |-  ( F : A --> B  ->  ran  ( 2nd  |`  F )  =  ran  F )
29 dffo2 5319 . 2  |-  ( ( 2nd  |`  F ) : F -onto-> ran  F  <->  ( ( 2nd  |`  F ) : F --> ran  F  /\  ran  ( 2nd  |`  F )  =  ran  F ) )
305, 28, 29sylanbrc 413 1  |-  ( F : A --> B  -> 
( 2nd  |`  F ) : F -onto-> ran  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316   E.wex 1453    e. wcel 1465    C_ wss 3041   <.cop 3500   ran crn 4510    |` cres 4511    Fn wfn 5088   -->wf 5089   -onto->wfo 5091   ` cfv 5093   2ndc2nd 6005
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fo 5099  df-fv 5101  df-2nd 6007
This theorem is referenced by:  f1o2ndf1  6093
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