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Theorem fo2ndf 5790
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 : -->  2nd  |`  F : F -onto-> ran  F

Proof of Theorem fo2ndf
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ffn 4989 . . . 4  F : -->  F  Fn
2 dffn3 4996 . . . 4  F  Fn  F :
--> ran  F
31, 2sylib 127 . . 3  F : -->  F : --> ran  F
4 f2ndf 5789 . . 3  F : --> ran  F  2nd  |`  F : F --> ran  F
53, 4syl 14 . 2  F : -->  2nd  |`  F : F --> ran  F
62, 4sylbi 114 . . . . 5  F  Fn  2nd  |`  F : F --> ran  F
71, 6syl 14 . . . 4  F : -->  2nd  |`  F : F --> ran  F
8 frn 4995 . . . 4  2nd  |`  F : F --> ran  F  ran  2nd  |`  F 
C_  ran  F
97, 8syl 14 . . 3  F : -->  ran  2nd  |`  F 
C_  ran  F
10 elrn2g 4468 . . . . . 6  ran  F  ran  F  <. ,  >.  F
1110ibi 165 . . . . 5  ran  F  <. , 
>.  F
12 fvres 5141 . . . . . . . . . 10  <. ,  >.  F  2nd  |`  F `
 <. , 
>.  2nd `  <. , 
>.
1312adantl 262 . . . . . . . . 9  F : -->  <. ,  >.  F  2nd  |`  F `  <. , 
>.  2nd `  <. , 
>.
14 vex 2554 . . . . . . . . . 10 
_V
15 vex 2554 . . . . . . . . . 10 
_V
1614, 15op2nd 5716 . . . . . . . . 9  2nd `  <. , 
>.
1713, 16syl6req 2086 . . . . . . . 8  F : -->  <. ,  >.  F  2nd  |`  F `
 <. , 
>.
18 f2ndf 5789 . . . . . . . . . 10  F : -->  2nd  |`  F : F -->
19 ffn 4989 . . . . . . . . . 10  2nd  |`  F : F -->  2nd  |`  F  Fn  F
2018, 19syl 14 . . . . . . . . 9  F : -->  2nd  |`  F  Fn  F
21 fnfvelrn 5242 . . . . . . . . 9  2nd  |`  F  Fn  F  <. ,  >.  F  2nd  |`  F `  <. ,  >.  ran  2nd  |`  F
2220, 21sylan 267 . . . . . . . 8  F : -->  <. ,  >.  F  2nd  |`  F `  <. , 
>.  ran  2nd  |`  F
2317, 22eqeltrd 2111 . . . . . . 7  F : -->  <. ,  >.  F  ran  2nd  |`  F
2423ex 108 . . . . . 6  F : -->  <. , 
>.  F  ran  2nd  |`  F
2524exlimdv 1697 . . . . 5  F : -->  <. ,  >.  F  ran  2nd  |`  F
2611, 25syl5 28 . . . 4  F : -->  ran  F  ran  2nd  |`  F
2726ssrdv 2945 . . 3  F : -->  ran 
F  C_  ran  2nd  |`  F
289, 27eqssd 2956 . 2  F : -->  ran  2nd  |`  F  ran  F
29 dffo2 5053 . 2  2nd  |`  F : F -onto-> ran  F  2nd  |`  F : F --> ran  F  ran  2nd  |`  F  ran  F
305, 28, 29sylanbrc 394 1  F : -->  2nd  |`  F : F -onto-> ran  F
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242  wex 1378   wcel 1390    C_ wss 2911   <.cop 3370   ran crn 4289    |` cres 4290    Fn wfn 4840   -->wf 4841   -onto->wfo 4843   ` cfv 4845   2ndc2nd 5708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-2nd 5710
This theorem is referenced by:  f1o2ndf1  5791
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