Theorem fvelrn 7074

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 2849	. . . . 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 6883	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
4	3	eleq1d 2846	. . . 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 7047	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
7		vex 3455	. . . . . 6 ⊢ 𝑥 ∈ V
8		opeq1 4833	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ⟨𝑦, (𝐹‘𝑥)⟩ = ⟨𝑥, (𝐹‘𝑥)⟩)
9	8	eleq1d 2846	. . . . . 6 ⊢ (𝑦 = 𝑥 → (⟨𝑦, (𝐹‘𝑥)⟩ ∈ 𝐹 ↔ ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹))
10	7, 9	spcev 3561	. . . . 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 6896	. . . . 5 ⊢ (𝐹‘𝑥) ∈ V
13	12	elrn2 5874	. . . 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 3518	. 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 2145  ⟨cop 4590  dom cdm 5651  ran crn 5652  Fun wfun 6531  ‘cfv 6537

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 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545

This theorem is used by:  nelrnfvne  7075  fnfvelrn  7078  eldmrexrn  7089  funfvima  7234  elunirn  7253  funeldmb  7367  rankwflemb  9793  rankwflembOLD  9794  dfac9  10208  fin1a2lem6  10476  ccatf1  14729  gsumpropd2lem  18861  nofv  28007  ltsres  28012  nolt02olem  28044  nosupno  28053  noinfno  28068  iedgedg  29621  usgredg3  29790  ushgredgedg  29803  ushgredgedgloop  29805  subgruhgredgd  29858  edginwlk  30208  iedginwlk  30210  cyclnumvtx  30381  opfv  33231  fnpreimac  33257  swrdrn2  33510  zartopn  34500  zarmxt1  34505  bj-elccinfty  38115  bj-minftyccb  38126  icoreunrn  38262  indexdom  38648  diaclN  42087  dia1elN  42091  docaclN  42161  dibclN  42199  sticksstones1  43176  dfac21  44052  harval3  44523  gneispace  45119  cncmpmax  46018  icccncfext  46866  stoweidlem27  47006  stoweidlem29  47008  stoweidlem59  47038  fourierdlem20  47106  fourierdlem63  47148  fourierdlem76  47161  fourierdlem82  47167  fourierdlem93  47178  fourierdlem113  47198  fge0iccico  47349  sge0sn  47358  sge0tsms  47359  sge0cl  47360  sge0isum  47406  hoicvr  47527  funressndmfvrn  48083  fcores  48106  afvelrn  48207  isubgredg  48933  gricushgr  48984  ushggricedg  48994  suppdm  49591