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Theorem fvelrn 7032
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 2826 . . . . 5 (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹𝐴 ∈ dom 𝐹))
21anbi2d 630 . . . 4 (𝑥 = 𝐴 → ((Fun 𝐹𝑥 ∈ dom 𝐹) ↔ (Fun 𝐹𝐴 ∈ dom 𝐹)))
3 fveq2 6847 . . . . 5 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
43eleq1d 2823 . . . 4 (𝑥 = 𝐴 → ((𝐹𝑥) ∈ ran 𝐹 ↔ (𝐹𝐴) ∈ ran 𝐹))
52, 4imbi12d 345 . . 3 (𝑥 = 𝐴 → (((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹) ↔ ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)))
6 funfvop 7005 . . . . 5 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
7 vex 3452 . . . . . 6 𝑥 ∈ V
8 opeq1 4835 . . . . . . 7 (𝑦 = 𝑥 → ⟨𝑦, (𝐹𝑥)⟩ = ⟨𝑥, (𝐹𝑥)⟩)
98eleq1d 2823 . . . . . 6 (𝑦 = 𝑥 → (⟨𝑦, (𝐹𝑥)⟩ ∈ 𝐹 ↔ ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹))
107, 9spcev 3568 . . . . 5 (⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹 → ∃𝑦𝑦, (𝐹𝑥)⟩ ∈ 𝐹)
116, 10syl 17 . . . 4 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ∃𝑦𝑦, (𝐹𝑥)⟩ ∈ 𝐹)
12 fvex 6860 . . . . 5 (𝐹𝑥) ∈ V
1312elrn2 5853 . . . 4 ((𝐹𝑥) ∈ ran 𝐹 ↔ ∃𝑦𝑦, (𝐹𝑥)⟩ ∈ 𝐹)
1411, 13sylibr 233 . . 3 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
155, 14vtoclg 3528 . 2 (𝐴 ∈ dom 𝐹 → ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹))
1615anabsi7 670 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  cop 4597  dom cdm 5638  ran crn 5639  Fun wfun 6495  cfv 6501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fn 6504  df-fv 6509
This theorem is referenced by:  nelrnfvne  7033  fnfvelrn  7036  eldmrexrn  7046  fvn0fvelrnOLD  7114  funfvima  7185  elunirn  7203  funeldmb  7309  rankwflemb  9736  dfac9  10079  fin1a2lem6  10348  gsumpropd2lem  18541  nofv  27021  sltres  27026  nolt02olem  27058  nosupno  27067  noinfno  27082  iedgedg  28043  usgredg3  28206  ushgredgedg  28219  ushgredgedgloop  28221  subgruhgredgd  28274  edginwlk  28625  iedginwlk  28627  opfv  31603  fnpreimac  31629  ccatf1  31847  swrdrn2  31850  zartopn  32496  zarmxt1  32501  bj-elccinfty  35714  bj-minftyccb  35725  icoreunrn  35859  indexdom  36222  diaclN  39542  dia1elN  39546  docaclN  39616  dibclN  39654  sticksstones1  40583  dfac21  41422  harval3  41884  gneispace  42480  cncmpmax  43311  icccncfext  44202  stoweidlem27  44342  stoweidlem29  44344  stoweidlem59  44374  fourierdlem20  44442  fourierdlem63  44484  fourierdlem76  44497  fourierdlem82  44503  fourierdlem93  44514  fourierdlem113  44534  fge0iccico  44685  sge0sn  44694  sge0tsms  44695  sge0cl  44696  sge0isum  44742  hoicvr  44863  funressndmfvrn  45352  fcores  45375  afvelrn  45474  isomushgr  46092  ushrisomgr  46107  suppdm  46665
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