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Theorem fvrnressn 6994
Description: If the value of a function is in the range of the function restricted to the singleton containing the argument, then the value of the function is in the range of the function. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
Assertion
Ref Expression
fvrnressn (𝑋𝑉 → ((𝐹𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹𝑋) ∈ ran 𝐹))

Proof of Theorem fvrnressn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-ima 5578 . . 3 (𝐹 “ {𝑋}) = ran (𝐹 ↾ {𝑋})
21eleq2i 2830 . 2 ((𝐹𝑋) ∈ (𝐹 “ {𝑋}) ↔ (𝐹𝑋) ∈ ran (𝐹 ↾ {𝑋}))
3 opeq1 4798 . . . . 5 (𝑥 = 𝑋 → ⟨𝑥, (𝐹𝑋)⟩ = ⟨𝑋, (𝐹𝑋)⟩)
43eleq1d 2823 . . . 4 (𝑥 = 𝑋 → (⟨𝑥, (𝐹𝑋)⟩ ∈ 𝐹 ↔ ⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹))
54spcegv 3524 . . 3 (𝑋𝑉 → (⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹 → ∃𝑥𝑥, (𝐹𝑋)⟩ ∈ 𝐹))
6 fvex 6748 . . . 4 (𝐹𝑋) ∈ V
7 elimasng 5970 . . . 4 ((𝑋𝑉 ∧ (𝐹𝑋) ∈ V) → ((𝐹𝑋) ∈ (𝐹 “ {𝑋}) ↔ ⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹))
86, 7mpan2 691 . . 3 (𝑋𝑉 → ((𝐹𝑋) ∈ (𝐹 “ {𝑋}) ↔ ⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹))
9 elrn2g 5773 . . . 4 ((𝐹𝑋) ∈ V → ((𝐹𝑋) ∈ ran 𝐹 ↔ ∃𝑥𝑥, (𝐹𝑋)⟩ ∈ 𝐹))
106, 9mp1i 13 . . 3 (𝑋𝑉 → ((𝐹𝑋) ∈ ran 𝐹 ↔ ∃𝑥𝑥, (𝐹𝑋)⟩ ∈ 𝐹))
115, 8, 103imtr4d 297 . 2 (𝑋𝑉 → ((𝐹𝑋) ∈ (𝐹 “ {𝑋}) → (𝐹𝑋) ∈ ran 𝐹))
122, 11syl5bir 246 1 (𝑋𝑉 → ((𝐹𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹𝑋) ∈ ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1543  wex 1787  wcel 2111  Vcvv 3420  {csn 4555  cop 4561  ran crn 5566  cres 5567  cima 5568  cfv 6397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5206  ax-nul 5213  ax-pr 5336
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3422  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-nul 4252  df-if 4454  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4834  df-br 5068  df-opab 5130  df-xp 5571  df-cnv 5573  df-dm 5575  df-rn 5576  df-res 5577  df-ima 5578  df-iota 6355  df-fv 6405
This theorem is referenced by:  fvn0fvelrn  6996  funressndmfvrn  44238
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