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Theorem ovelrn 5572
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 5569 . . 3  F  Fn  X.  ran  F  {  |  F }
21eleq2d 2089 . 2  F  Fn  X.  C  ran  F  C 
{  |  F }
3 elex 2543 . . . 4  C  {  |  F }  C  _V
43a1i 9 . . 3  F  Fn  X.  C  {  |  F }  C  _V
5 fnovex 5462 . . . . . 6  F  Fn  X.  F  _V
6 eleq1 2082 . . . . . 6  C  F  C  _V  F 
_V
75, 6syl5ibrcom 146 . . . . 5  F  Fn  X.  C  F  C  _V
873expb 1091 . . . 4  F  Fn  X.  C  F 
C  _V
98rexlimdvva 2418 . . 3  F  Fn  X.  C  F 
C  _V
10 eqeq1 2028 . . . . . 6  C  F  C  F
11102rexbidv 2327 . . . . 5  C  F  C  F
1211elabg 2665 . . . 4  C  _V  C  {  |  F }  C  F
1312a1i 9 . . 3  F  Fn  X.  C  _V  C  {  |  F }  C  F
144, 9, 13pm5.21ndd 608 . 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 873   wceq 1228   wcel 1374   {cab 2008  wrex 2285   _Vcvv 2535    X. cxp 4270   ran crn 4273    Fn wfn 4824  (class class class)co 5436
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837  df-ov 5439
This theorem is referenced by: (None)
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