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Theorem fo2ndf 5899
Description: The  2nd (second member of an ordered pair) function restricted to a function  F is a function of  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 5097 . . . 4  |-  ( F : A --> B  ->  F  Fn  A )
2 dffn3 5104 . . . 4  |-  ( F  Fn  A  <->  F : A
--> ran  F )
31, 2sylib 120 . . 3  |-  ( F : A --> B  ->  F : A --> ran  F
)
4 f2ndf 5898 . . 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 5103 . . . 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 4573 . . . . . 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 5250 . . . . . . . . . 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 2613 . . . . . . . . . 10  |-  x  e. 
_V
15 vex 2613 . . . . . . . . . 10  |-  y  e. 
_V
1614, 15op2nd 5825 . . . . . . . . 9  |-  ( 2nd `  <. x ,  y
>. )  =  y
1713, 16syl6req 2132 . . . . . . . 8  |-  ( ( F : A --> B  /\  <.
x ,  y >.  e.  F )  ->  y  =  ( ( 2nd  |`  F ) `  <. x ,  y >. )
)
18 f2ndf 5898 . . . . . . . . . 10  |-  ( F : A --> B  -> 
( 2nd  |`  F ) : F --> B )
19 ffn 5097 . . . . . . . . . 10  |-  ( ( 2nd  |`  F ) : F --> B  ->  ( 2nd  |`  F )  Fn  F )
2018, 19syl 14 . . . . . . . . 9  |-  ( F : A --> B  -> 
( 2nd  |`  F )  Fn  F )
21 fnfvelrn 5351 . . . . . . . . 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 2159 . . . . . . 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 1742 . . . . 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 3014 . . 3  |-  ( F : A --> B  ->  ran  F  C_  ran  ( 2nd  |`  F ) )
289, 27eqssd 3025 . 2  |-  ( F : A --> B  ->  ran  ( 2nd  |`  F )  =  ran  F )
29 dffo2 5161 . 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 1285   E.wex 1422    e. wcel 1434    C_ wss 2982   <.cop 3419   ran crn 4392    |` cres 4393    Fn wfn 4947   -->wf 4948   -onto->wfo 4950   ` cfv 4952   2ndc2nd 5817
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-fo 4958  df-fv 4960  df-2nd 5819
This theorem is referenced by:  f1o2ndf1  5900
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