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Theorem fvrnressn 6915
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 5561 . . 3 (𝐹 “ {𝑋}) = ran (𝐹 ↾ {𝑋})
21eleq2i 2901 . 2 ((𝐹𝑋) ∈ (𝐹 “ {𝑋}) ↔ (𝐹𝑋) ∈ ran (𝐹 ↾ {𝑋}))
3 opeq1 4795 . . . . 5 (𝑥 = 𝑋 → ⟨𝑥, (𝐹𝑋)⟩ = ⟨𝑋, (𝐹𝑋)⟩)
43eleq1d 2894 . . . 4 (𝑥 = 𝑋 → (⟨𝑥, (𝐹𝑋)⟩ ∈ 𝐹 ↔ ⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹))
54spcegv 3594 . . 3 (𝑋𝑉 → (⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹 → ∃𝑥𝑥, (𝐹𝑋)⟩ ∈ 𝐹))
6 fvex 6676 . . . 4 (𝐹𝑋) ∈ V
7 elimasng 5948 . . . 4 ((𝑋𝑉 ∧ (𝐹𝑋) ∈ V) → ((𝐹𝑋) ∈ (𝐹 “ {𝑋}) ↔ ⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹))
86, 7mpan2 687 . . 3 (𝑋𝑉 → ((𝐹𝑋) ∈ (𝐹 “ {𝑋}) ↔ ⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹))
9 elrn2g 5754 . . . 4 ((𝐹𝑋) ∈ V → ((𝐹𝑋) ∈ ran 𝐹 ↔ ∃𝑥𝑥, (𝐹𝑋)⟩ ∈ 𝐹))
106, 9mp1i 13 . . 3 (𝑋𝑉 → ((𝐹𝑋) ∈ ran 𝐹 ↔ ∃𝑥𝑥, (𝐹𝑋)⟩ ∈ 𝐹))
115, 8, 103imtr4d 295 . 2 (𝑋𝑉 → ((𝐹𝑋) ∈ (𝐹 “ {𝑋}) → (𝐹𝑋) ∈ ran 𝐹))
122, 11syl5bir 244 1 (𝑋𝑉 → ((𝐹𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹𝑋) ∈ ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wex 1771  wcel 2105  Vcvv 3492  {csn 4557  cop 4563  ran crn 5549  cres 5550  cima 5551  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fv 6356
This theorem is referenced by:  fvn0fvelrn  6917  funressndmfvrn  43156
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