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Theorem ovelrn 5591
Description: A member of an operation's range is a value of the operation. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)
Assertion
Ref Expression
ovelrn  F  Fn  X.  C  ran  F  C  F
Distinct variable groups:   ,,   ,,   , C,   , F,

Proof of Theorem ovelrn
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fnrnov 5588 . . 3  F  Fn  X.  ran  F  {  |  F }
21eleq2d 2104 . 2  F  Fn  X.  C  ran  F  C 
{  |  F }
3 elex 2560 . . . 4  C  {  |  F }  C  _V
43a1i 9 . . 3  F  Fn  X.  C  {  |  F }  C  _V
5 fnovex 5481 . . . . . 6  F  Fn  X.  F  _V
6 eleq1 2097 . . . . . 6  C  F  C  _V  F 
_V
75, 6syl5ibrcom 146 . . . . 5  F  Fn  X.  C  F  C  _V
873expb 1104 . . . 4  F  Fn  X.  C  F 
C  _V
98rexlimdvva 2434 . . 3  F  Fn  X.  C  F 
C  _V
10 eqeq1 2043 . . . . . 6  C  F  C  F
11102rexbidv 2343 . . . . 5  C  F  C  F
1211elabg 2682 . . . 4  C  _V  C  {  |  F }  C  F
1312a1i 9 . . 3  F  Fn  X.  C  _V  C  {  |  F }  C  F
144, 9, 13pm5.21ndd 620 . 2  F  Fn  X.  C  {  |  F }  C  F
152, 14bitrd 177 1  F  Fn  X.  C  ran  F  C  F
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98   w3a 884   wceq 1242   wcel 1390   {cab 2023  wrex 2301   _Vcvv 2551    X. cxp 4286   ran crn 4289    Fn wfn 4840  (class class class)co 5455
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-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
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-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-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853  df-ov 5458
This theorem is referenced by: (None)
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