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Theorem elrnmpt1 4528
Description: Elementhood in an image set. (Contributed by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
rnmpt.1  F  |->
Assertion
Ref Expression
elrnmpt1  V  ran  F

Proof of Theorem elrnmpt1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . 4 
_V
2 id 19 . . . . . . 7
3 csbeq1a 2854 . . . . . . 7  [_  ]_
42, 3eleq12d 2105 . . . . . 6 
[_  ]_
5 csbeq1a 2854 . . . . . . 7  [_  ]_
65biantrud 288 . . . . . 6  [_  ]_  [_  ]_  [_  ]_
74, 6bitr2d 178 . . . . 5  [_  ]_  [_  ]_
87equcoms 1591 . . . 4  [_  ]_  [_  ]_
91, 8spcev 2641 . . 3  [_  ]_  [_  ]_
10 df-rex 2306 . . . . . 6
11 nfv 1418 . . . . . . 7  F/
12 nfcsb1v 2876 . . . . . . . . 9  F/_ [_  ]_
1312nfcri 2169 . . . . . . . 8  F/  [_  ]_
14 nfcsb1v 2876 . . . . . . . . 9  F/_ [_  ]_
1514nfeq2 2186 . . . . . . . 8  F/  [_  ]_
1613, 15nfan 1454 . . . . . . 7  F/  [_  ]_  [_  ]_
175eqeq2d 2048 . . . . . . . 8 
[_  ]_
184, 17anbi12d 442 . . . . . . 7  [_  ]_  [_  ]_
1911, 16, 18cbvex 1636 . . . . . 6  [_  ]_  [_  ]_
2010, 19bitri 173 . . . . 5  [_  ]_  [_  ]_
21 eqeq1 2043 . . . . . . 7  [_  ]_  [_  ]_
2221anbi2d 437 . . . . . 6  [_  ]_  [_  ]_  [_  ]_  [_  ]_
2322exbidv 1703 . . . . 5 
[_  ]_ 
[_  ]_  [_  ]_  [_  ]_
2420, 23syl5bb 181 . . . 4  [_  ]_  [_  ]_
25 rnmpt.1 . . . . 5  F  |->
2625rnmpt 4525 . . . 4  ran  F  {  |  }
2724, 26elab2g 2683 . . 3  V  ran  F  [_  ]_  [_  ]_
289, 27syl5ibr 145 . 2  V  ran  F
2928impcom 116 1  V  ran  F
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242  wex 1378   wcel 1390  wrex 2301   [_csb 2846    |-> 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-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-br 3756  df-opab 3810  df-mpt 3811  df-cnv 4296  df-dm 4298  df-rn 4299
This theorem is referenced by:  fliftel1  5377
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