Theorem fvelrn 7075

Description: A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)

Assertion

Ref	Expression
fvelrn	⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹)

Proof of Theorem fvelrn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2853	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹 ↔ 𝐴 ∈ dom 𝐹))
2	1	anbi2d 642	. . . 4 ⊢ (𝑥 = 𝐴 → ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) ↔ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)))
3		fveq2 6885	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
4	3	eleq1d 2850	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) ∈ ran 𝐹 ↔ (𝐹‘𝐴) ∈ ran 𝐹))
5	2, 4	imbi12d 347	. . 3 ⊢ (𝑥 = 𝐴 → (((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) ↔ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹)))
6		funfvop 7049	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
7		vex 3461	. . . . . 6 ⊢ 𝑥 ∈ V
8		opeq1 4840	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ⟨𝑦, (𝐹‘𝑥)⟩ = ⟨𝑥, (𝐹‘𝑥)⟩)
9	8	eleq1d 2850	. . . . . 6 ⊢ (𝑦 = 𝑥 → (⟨𝑦, (𝐹‘𝑥)⟩ ∈ 𝐹 ↔ ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹))
10	7, 9	spcev 3567	. . . . 5 ⊢ (⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹 → ∃𝑦⟨𝑦, (𝐹‘𝑥)⟩ ∈ 𝐹)
11	6, 10	syl 18	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ∃𝑦⟨𝑦, (𝐹‘𝑥)⟩ ∈ 𝐹)
12		fvex 6898	. . . . 5 ⊢ (𝐹‘𝑥) ∈ V
13	12	elrn2 5884	. . . 4 ⊢ ((𝐹‘𝑥) ∈ ran 𝐹 ↔ ∃𝑦⟨𝑦, (𝐹‘𝑥)⟩ ∈ 𝐹)
14	11, 13	sylibr 237	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹)
15	5, 14	vtoclg 3524	. 2 ⊢ (𝐴 ∈ dom 𝐹 → ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹))
16	15	anabsi7 684	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ⟨cop 4597  dom cdm 5663  ran crn 5664  Fun wfun 6534  ‘cfv 6540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fn 6543  df-fv 6548

This theorem is used by:  nelrnfvne  7076  fnfvelrn  7079  eldmrexrn  7090  funfvima  7235  elunirn  7254  funeldmb  7368  rankwflemb  9772  dfac9  10136  fin1a2lem6  10404  ccatf1  14646  gsumpropd2lem  18769  nofv  27872  ltsres  27877  nolt02olem  27909  nosupno  27918  noinfno  27933  iedgedg  29455  usgredg3  29624  ushgredgedg  29637  ushgredgedgloop  29639  subgruhgredgd  29692  edginwlk  30042  iedginwlk  30044  cyclnumvtx  30215  opfv  33060  fnpreimac  33086  swrdrn2  33340  zartopn  34329  zarmxt1  34334  bj-elccinfty  37915  bj-minftyccb  37926  icoreunrn  38062  indexdom  38443  diaclN  41882  dia1elN  41886  docaclN  41956  dibclN  41994  sticksstones1  42971  dfac21  43851  harval3  44322  gneispace  44918  cncmpmax  45810  icccncfext  46659  stoweidlem27  46799  stoweidlem29  46801  stoweidlem59  46831  fourierdlem20  46899  fourierdlem63  46941  fourierdlem76  46954  fourierdlem82  46960  fourierdlem93  46971  fourierdlem113  46991  fge0iccico  47142  sge0sn  47151  sge0tsms  47152  sge0cl  47153  sge0isum  47199  hoicvr  47320  funressndmfvrn  47839  fcores  47862  afvelrn  47963  isubgredg  48689  gricushgr  48740  ushggricedg  48750  suppdm  49347