Theorem fvelrn 6954

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 2826	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹 ↔ 𝐴 ∈ dom 𝐹))
2	1	anbi2d 629	. . . 4 ⊢ (𝑥 = 𝐴 → ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) ↔ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)))
3		fveq2 6774	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
4	3	eleq1d 2823	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) ∈ ran 𝐹 ↔ (𝐹‘𝐴) ∈ ran 𝐹))
5	2, 4	imbi12d 345	. . 3 ⊢ (𝑥 = 𝐴 → (((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) ↔ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹)))
6		funfvop 6927	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
7		vex 3436	. . . . . 6 ⊢ 𝑥 ∈ V
8		opeq1 4804	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ⟨𝑦, (𝐹‘𝑥)⟩ = ⟨𝑥, (𝐹‘𝑥)⟩)
9	8	eleq1d 2823	. . . . . 6 ⊢ (𝑦 = 𝑥 → (⟨𝑦, (𝐹‘𝑥)⟩ ∈ 𝐹 ↔ ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹))
10	7, 9	spcev 3545	. . . . 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 6787	. . . . 5 ⊢ (𝐹‘𝑥) ∈ V
13	12	elrn2 5801	. . . 4 ⊢ ((𝐹‘𝑥) ∈ ran 𝐹 ↔ ∃𝑦⟨𝑦, (𝐹‘𝑥)⟩ ∈ 𝐹)
14	11, 13	sylibr 233	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹)
15	5, 14	vtoclg 3505	. 2 ⊢ (𝐴 ∈ dom 𝐹 → ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹))
16	15	anabsi7 668	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ⟨cop 4567  dom cdm 5589  ran crn 5590  Fun wfun 6427  ‘cfv 6433

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441

This theorem is referenced by:  nelrnfvne  6955  fnfvelrn  6958  eldmrexrn  6967  fvn0fvelrn  7035  funfvima  7106  elunirn  7124  rankwflemb  9551  dfac9  9892  fin1a2lem6  10161  gsumpropd2lem  18363  iedgedg  27420  usgredg3  27583  ushgredgedg  27596  ushgredgedgloop  27598  subgruhgredgd  27651  edginwlk  28002  iedginwlk  28004  opfv  30982  fnpreimac  31008  ccatf1  31223  swrdrn2  31226  zartopn  31825  zarmxt1  31830  funeldmb  33737  nofv  33860  sltres  33865  nolt02olem  33897  nosupno  33906  noinfno  33921  bj-elccinfty  35385  bj-minftyccb  35396  icoreunrn  35530  indexdom  35892  diaclN  39064  dia1elN  39068  docaclN  39138  dibclN  39176  sticksstones1  40102  dfac21  40891  harval3  41145  gneispace  41744  cncmpmax  42575  icccncfext  43428  stoweidlem27  43568  stoweidlem29  43570  stoweidlem59  43600  fourierdlem20  43668  fourierdlem63  43710  fourierdlem76  43723  fourierdlem82  43729  fourierdlem93  43740  fourierdlem113  43760  fge0iccico  43908  sge0sn  43917  sge0tsms  43918  sge0cl  43919  sge0isum  43965  hoicvr  44086  funressndmfvrn  44538  fcores  44561  afvelrn  44660  isomushgr  45278  ushrisomgr  45293  suppdm  45851