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Theorem fvrnressn 7181
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 5698 . . 3 (𝐹 “ {𝑋}) = ran (𝐹 ↾ {𝑋})
21eleq2i 2833 . 2 ((𝐹𝑋) ∈ (𝐹 “ {𝑋}) ↔ (𝐹𝑋) ∈ ran (𝐹 ↾ {𝑋}))
3 opeq1 4873 . . . . 5 (𝑥 = 𝑋 → ⟨𝑥, (𝐹𝑋)⟩ = ⟨𝑋, (𝐹𝑋)⟩)
43eleq1d 2826 . . . 4 (𝑥 = 𝑋 → (⟨𝑥, (𝐹𝑋)⟩ ∈ 𝐹 ↔ ⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹))
54spcegv 3597 . . 3 (𝑋𝑉 → (⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹 → ∃𝑥𝑥, (𝐹𝑋)⟩ ∈ 𝐹))
6 fvex 6919 . . . 4 (𝐹𝑋) ∈ V
7 elimasng 6107 . . . 4 ((𝑋𝑉 ∧ (𝐹𝑋) ∈ V) → ((𝐹𝑋) ∈ (𝐹 “ {𝑋}) ↔ ⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹))
86, 7mpan2 691 . . 3 (𝑋𝑉 → ((𝐹𝑋) ∈ (𝐹 “ {𝑋}) ↔ ⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹))
9 elrn2g 5901 . . . 4 ((𝐹𝑋) ∈ V → ((𝐹𝑋) ∈ ran 𝐹 ↔ ∃𝑥𝑥, (𝐹𝑋)⟩ ∈ 𝐹))
106, 9mp1i 13 . . 3 (𝑋𝑉 → ((𝐹𝑋) ∈ ran 𝐹 ↔ ∃𝑥𝑥, (𝐹𝑋)⟩ ∈ 𝐹))
115, 8, 103imtr4d 294 . 2 (𝑋𝑉 → ((𝐹𝑋) ∈ (𝐹 “ {𝑋}) → (𝐹𝑋) ∈ ran 𝐹))
122, 11biimtrrid 243 1 (𝑋𝑉 → ((𝐹𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹𝑋) ∈ ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  {csn 4626  cop 4632  ran crn 5686  cres 5687  cima 5688  cfv 6561
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fv 6569
This theorem is referenced by:  fvn0fvelrnOLD  7183  funressndmfvrn  47056
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