Theorem funressndmfvrn 47040

Description: The value of a function 𝐹 at a set 𝐴 is in the range of the function 𝐹 if 𝐴 is in the domain of the function 𝐹. It is sufficient that 𝐹 is a function at 𝐴. (Contributed by AV, 1-Sep-2022.)

Assertion

Ref	Expression
funressndmfvrn	⊢ ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹)

Proof of Theorem funressndmfvrn

Step	Hyp	Ref	Expression
1		simpr 484	. 2 ⊢ ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → 𝐴 ∈ dom 𝐹)
2		fvressn 7157	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 → ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹‘𝐴))
3	2	adantl 481	. . 3 ⊢ ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹‘𝐴))
4		eldmressnsn 6016	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ dom (𝐹 ↾ {𝐴}))
5		fvelrn 7071	. . . 4 ⊢ ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom (𝐹 ↾ {𝐴})) → ((𝐹 ↾ {𝐴})‘𝐴) ∈ ran (𝐹 ↾ {𝐴}))
6	4, 5	sylan2 593	. . 3 ⊢ ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → ((𝐹 ↾ {𝐴})‘𝐴) ∈ ran (𝐹 ↾ {𝐴}))
7	3, 6	eqeltrrd 2836	. 2 ⊢ ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran (𝐹 ↾ {𝐴}))
8		fvrnressn 7156	. 2 ⊢ (𝐴 ∈ dom 𝐹 → ((𝐹‘𝐴) ∈ ran (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) ∈ ran 𝐹))
9	1, 7, 8	sylc 65	1 ⊢ ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {csn 4606  dom cdm 5659  ran crn 5660   ↾ cres 5661  Fun wfun 6530  ‘cfv 6536

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 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-fv 6544

This theorem is referenced by:  dfatelrn  47127