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Theorem fvelrn 7096
Description: A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)
Assertion
Ref Expression
fvelrn ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)

Proof of Theorem fvelrn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2827 . . . . 5 (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹𝐴 ∈ dom 𝐹))
21anbi2d 630 . . . 4 (𝑥 = 𝐴 → ((Fun 𝐹𝑥 ∈ dom 𝐹) ↔ (Fun 𝐹𝐴 ∈ dom 𝐹)))
3 fveq2 6907 . . . . 5 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
43eleq1d 2824 . . . 4 (𝑥 = 𝐴 → ((𝐹𝑥) ∈ ran 𝐹 ↔ (𝐹𝐴) ∈ ran 𝐹))
52, 4imbi12d 344 . . 3 (𝑥 = 𝐴 → (((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹) ↔ ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)))
6 funfvop 7070 . . . . 5 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
7 vex 3482 . . . . . 6 𝑥 ∈ V
8 opeq1 4878 . . . . . . 7 (𝑦 = 𝑥 → ⟨𝑦, (𝐹𝑥)⟩ = ⟨𝑥, (𝐹𝑥)⟩)
98eleq1d 2824 . . . . . 6 (𝑦 = 𝑥 → (⟨𝑦, (𝐹𝑥)⟩ ∈ 𝐹 ↔ ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹))
107, 9spcev 3606 . . . . 5 (⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹 → ∃𝑦𝑦, (𝐹𝑥)⟩ ∈ 𝐹)
116, 10syl 17 . . . 4 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ∃𝑦𝑦, (𝐹𝑥)⟩ ∈ 𝐹)
12 fvex 6920 . . . . 5 (𝐹𝑥) ∈ V
1312elrn2 5906 . . . 4 ((𝐹𝑥) ∈ ran 𝐹 ↔ ∃𝑦𝑦, (𝐹𝑥)⟩ ∈ 𝐹)
1411, 13sylibr 234 . . 3 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
155, 14vtoclg 3554 . 2 (𝐴 ∈ dom 𝐹 → ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹))
1615anabsi7 671 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  cop 4637  dom cdm 5689  ran crn 5690  Fun wfun 6557  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by:  nelrnfvne  7097  fnfvelrn  7100  eldmrexrn  7111  fvn0fvelrnOLD  7183  funfvima  7250  elunirn  7271  funeldmb  7379  rankwflemb  9831  dfac9  10175  fin1a2lem6  10443  gsumpropd2lem  18705  nofv  27717  sltres  27722  nolt02olem  27754  nosupno  27763  noinfno  27778  iedgedg  29082  usgredg3  29248  ushgredgedg  29261  ushgredgedgloop  29263  subgruhgredgd  29316  edginwlk  29668  iedginwlk  29670  opfv  32661  fnpreimac  32688  ccatf1  32918  swrdrn2  32924  zartopn  33836  zarmxt1  33841  bj-elccinfty  37197  bj-minftyccb  37208  icoreunrn  37342  indexdom  37721  diaclN  41033  dia1elN  41037  docaclN  41107  dibclN  41145  sticksstones1  42128  dfac21  43055  harval3  43528  gneispace  44124  cncmpmax  44970  icccncfext  45843  stoweidlem27  45983  stoweidlem29  45985  stoweidlem59  46015  fourierdlem20  46083  fourierdlem63  46125  fourierdlem76  46138  fourierdlem82  46144  fourierdlem93  46155  fourierdlem113  46175  fge0iccico  46326  sge0sn  46335  sge0tsms  46336  sge0cl  46337  sge0isum  46383  hoicvr  46504  funressndmfvrn  46994  fcores  47017  afvelrn  47118  isubgredg  47790  gricushgr  47824  ushggricedg  47834  suppdm  48356
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