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Theorem funressndmfvrn 48030
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) → (𝐹𝐴) ∈ ran 𝐹)

Proof of Theorem funressndmfvrn
StepHypRef Expression
1 simpr 490 . 2 ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → 𝐴 ∈ dom 𝐹)
2 fvressn 7155 . . . 4 (𝐴 ∈ dom 𝐹 → ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹𝐴))
32adantl 487 . . 3 ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹𝐴))
4 eldmressnsn 6012 . . . 4 (𝐴 ∈ dom 𝐹𝐴 ∈ dom (𝐹 ↾ {𝐴}))
5 fvelrn 7065 . . . 4 ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom (𝐹 ↾ {𝐴})) → ((𝐹 ↾ {𝐴})‘𝐴) ∈ ran (𝐹 ↾ {𝐴}))
64, 5sylan2 605 . . 3 ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → ((𝐹 ↾ {𝐴})‘𝐴) ∈ ran (𝐹 ↾ {𝐴}))
73, 6eqeltrrd 2861 . 2 ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran (𝐹 ↾ {𝐴}))
8 fvrnressn 7154 . 2 (𝐴 ∈ dom 𝐹 → ((𝐹𝐴) ∈ ran (𝐹 ↾ {𝐴}) → (𝐹𝐴) ∈ ran 𝐹))
91, 7, 8sylc 66 1 ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)
Colors of variables:    wff setvar class
This proof depends on syntax axioms:  wi 4  wa 401   = wceq 1570  wcel 2145  {csn 4584  dom cdm 5648  ran crn 5649  cres 5650  Fun wfun 6522  cfv 6528
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2732  ax-sep 5249  ax-nul 5260  ax-pr 5391
This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6484  df-fun 6530  df-fn 6531  df-fv 6536
This theorem is used by:  dfatelrn  48117
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