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Theorem fvelrn 5241
Description: A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)
Assertion
Ref Expression
fvelrn  Fun  F  dom  F  F `  ran  F

Proof of Theorem fvelrn
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2097 . . . . 5  dom  F  dom  F
21anbi2d 437 . . . 4  Fun  F  dom  F  Fun 
F  dom  F
3 fveq2 5121 . . . . 5  F `  F `
43eleq1d 2103 . . . 4  F `  ran  F  F `

ran  F
52, 4imbi12d 223 . . 3  Fun  F  dom  F  F `  ran  F  Fun  F  dom  F  F `  ran  F
6 funfvop 5222 . . . . 5  Fun  F  dom  F  <. ,  F `
 >.  F
7 vex 2554 . . . . . 6 
_V
8 opeq1 3540 . . . . . . 7  <. ,  F `  >.  <. ,  F `  >.
98eleq1d 2103 . . . . . 6  <. ,  F `
 >.  F  <. ,  F ` 
>.  F
107, 9spcev 2641 . . . . 5  <. ,  F `  >.  F  <. ,  F `  >.  F
116, 10syl 14 . . . 4  Fun  F  dom  F  <. ,  F ` 
>.  F
12 funfvex 5135 . . . . 5  Fun  F  dom  F  F `  _V
13 elrn2g 4468 . . . . 5  F `  _V  F `  ran  F  <. ,  F `
 >.  F
1412, 13syl 14 . . . 4  Fun  F  dom  F  F `  ran  F  <. ,  F `
 >.  F
1511, 14mpbird 156 . . 3  Fun  F  dom  F  F `  ran  F
165, 15vtoclg 2607 . 2  dom  F  Fun  F  dom  F  F `  ran  F
1716anabsi7 515 1  Fun  F  dom  F  F `  ran  F
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242  wex 1378   wcel 1390   _Vcvv 2551   <.cop 3370   dom cdm 4288   ran crn 4289   Fun wfun 4839   ` cfv 4845
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-sbc 2759  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-br 3756  df-opab 3810  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
This theorem is referenced by:  fnfvelrn  5242  eldmrexrn  5251  funfvima  5333  elunirn  5348
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