Theorem funressndmfvrn 46349

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 7165	. . . 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 6022	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ dom (𝐹 ↾ {𝐴}))
5		fvelrn 7080	. . . 4 ⊢ ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom (𝐹 ↾ {𝐴})) → ((𝐹 ↾ {𝐴})‘𝐴) ∈ ran (𝐹 ↾ {𝐴}))
6	4, 5	sylan2 592	. . 3 ⊢ ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → ((𝐹 ↾ {𝐴})‘𝐴) ∈ ran (𝐹 ↾ {𝐴}))
7	3, 6	eqeltrrd 2829	. 2 ⊢ ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran (𝐹 ↾ {𝐴}))
8		fvrnressn 7164	. 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 1534   ∈ wcel 2099  {csn 4624  dom cdm 5672  ran crn 5673   ↾ cres 5674  Fun wfun 6536  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550

This theorem is referenced by:  dfatelrn  46434