Theorem fvelrn 7030

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 2816	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹 ↔ 𝐴 ∈ dom 𝐹))
2	1	anbi2d 630	. . . 4 ⊢ (𝑥 = 𝐴 → ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) ↔ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)))
3		fveq2 6840	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
4	3	eleq1d 2813	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) ∈ ran 𝐹 ↔ (𝐹‘𝐴) ∈ ran 𝐹))
5	2, 4	imbi12d 344	. . 3 ⊢ (𝑥 = 𝐴 → (((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) ↔ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹)))
6		funfvop 7004	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
7		vex 3448	. . . . . 6 ⊢ 𝑥 ∈ V
8		opeq1 4833	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ⟨𝑦, (𝐹‘𝑥)⟩ = ⟨𝑥, (𝐹‘𝑥)⟩)
9	8	eleq1d 2813	. . . . . 6 ⊢ (𝑦 = 𝑥 → (⟨𝑦, (𝐹‘𝑥)⟩ ∈ 𝐹 ↔ ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹))
10	7, 9	spcev 3569	. . . . 5 ⊢ (⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹 → ∃𝑦⟨𝑦, (𝐹‘𝑥)⟩ ∈ 𝐹)
11	6, 10	syl 17	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ∃𝑦⟨𝑦, (𝐹‘𝑥)⟩ ∈ 𝐹)
12		fvex 6853	. . . . 5 ⊢ (𝐹‘𝑥) ∈ V
13	12	elrn2 5846	. . . 4 ⊢ ((𝐹‘𝑥) ∈ ran 𝐹 ↔ ∃𝑦⟨𝑦, (𝐹‘𝑥)⟩ ∈ 𝐹)
14	11, 13	sylibr 234	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹)
15	5, 14	vtoclg 3517	. 2 ⊢ (𝐴 ∈ dom 𝐹 → ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹))
16	15	anabsi7 671	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ⟨cop 4591  dom cdm 5631  ran crn 5632  Fun wfun 6493  ‘cfv 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6452  df-fun 6501  df-fn 6502  df-fv 6507

This theorem is referenced by:  nelrnfvne  7031  fnfvelrn  7034  eldmrexrn  7045  fvn0fvelrnOLD  7117  funfvima  7186  elunirn  7207  funeldmb  7316  rankwflemb  9722  dfac9  10066  fin1a2lem6  10334  gsumpropd2lem  18582  nofv  27545  sltres  27550  nolt02olem  27582  nosupno  27591  noinfno  27606  iedgedg  28953  usgredg3  29119  ushgredgedg  29132  ushgredgedgloop  29134  subgruhgredgd  29187  edginwlk  29538  iedginwlk  29540  cyclnumvtx  29703  opfv  32541  fnpreimac  32568  ccatf1  32843  swrdrn2  32849  zartopn  33838  zarmxt1  33843  bj-elccinfty  37175  bj-minftyccb  37186  icoreunrn  37320  indexdom  37701  diaclN  41017  dia1elN  41021  docaclN  41091  dibclN  41129  sticksstones1  42107  dfac21  43028  harval3  43500  gneispace  44096  cncmpmax  44999  icccncfext  45858  stoweidlem27  45998  stoweidlem29  46000  stoweidlem59  46030  fourierdlem20  46098  fourierdlem63  46140  fourierdlem76  46153  fourierdlem82  46159  fourierdlem93  46170  fourierdlem113  46190  fge0iccico  46341  sge0sn  46350  sge0tsms  46351  sge0cl  46352  sge0isum  46398  hoicvr  46519  funressndmfvrn  47018  fcores  47041  afvelrn  47142  isubgredg  47839  gricushgr  47890  ushggricedg  47900  suppdm  48472