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Theorem elrnmptg 4529
Description: Membership in the range of a function. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
rnmpt.1  F  |->
Assertion
Ref Expression
elrnmptg  V  C  ran  F  C
Distinct variable group:   , C
Allowed substitution hints:   ()   ()    F()    V()

Proof of Theorem elrnmptg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 rnmpt.1 . . . 4  F  |->
21rnmpt 4525 . . 3  ran  F  {  |  }
32eleq2i 2101 . 2  C  ran  F  C 
{  |  }
4 r19.29 2444 . . . . 5  V  C  V  C
5 eleq1 2097 . . . . . . . 8  C  C  V  V
65biimparc 283 . . . . . . 7  V  C  C  V
7 elex 2560 . . . . . . 7  C  V  C  _V
86, 7syl 14 . . . . . 6  V  C  C  _V
98rexlimivw 2423 . . . . 5  V  C  C  _V
104, 9syl 14 . . . 4  V  C  C  _V
1110ex 108 . . 3  V  C  C  _V
12 eqeq1 2043 . . . . 5  C  C
1312rexbidv 2321 . . . 4  C  C
1413elab3g 2687 . . 3  C  C  _V  C  {  |  }  C
1511, 14syl 14 . 2  V  C  {  |  }  C
163, 15syl5bb 181 1  V  C  ran  F  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wcel 1390   {cab 2023  wral 2300  wrex 2301   _Vcvv 2551    |-> cmpt 3809   ran crn 4289
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-bnd 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-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-mpt 3811  df-cnv 4296  df-dm 4298  df-rn 4299
This theorem is referenced by:  elrnmpti  4530  fliftel  5376
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