Theorem funressndmfvrn 47433

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 7119	. . . 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 5993	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ dom (𝐹 ↾ {𝐴}))
5		fvelrn 7032	. . . 4 ⊢ ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom (𝐹 ↾ {𝐴})) → ((𝐹 ↾ {𝐴})‘𝐴) ∈ ran (𝐹 ↾ {𝐴}))
6	4, 5	sylan2 594	. . 3 ⊢ ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → ((𝐹 ↾ {𝐴})‘𝐴) ∈ ran (𝐹 ↾ {𝐴}))
7	3, 6	eqeltrrd 2838	. 2 ⊢ ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran (𝐹 ↾ {𝐴}))
8		fvrnressn 7118	. 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 1542   ∈ wcel 2114  {csn 4582  dom cdm 5634  ran crn 5635   ↾ cres 5636  Fun wfun 6496  ‘cfv 6502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-fv 6510

This theorem is referenced by:  dfatelrn  47520