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Theorem fvrnressn 6656
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 5325 . . 3 (𝐹 “ {𝑋}) = ran (𝐹 ↾ {𝑋})
21eleq2i 2870 . 2 ((𝐹𝑋) ∈ (𝐹 “ {𝑋}) ↔ (𝐹𝑋) ∈ ran (𝐹 ↾ {𝑋}))
3 opeq1 4593 . . . . 5 (𝑥 = 𝑋 → ⟨𝑥, (𝐹𝑋)⟩ = ⟨𝑋, (𝐹𝑋)⟩)
43eleq1d 2863 . . . 4 (𝑥 = 𝑋 → (⟨𝑥, (𝐹𝑋)⟩ ∈ 𝐹 ↔ ⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹))
54spcegv 3482 . . 3 (𝑋𝑉 → (⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹 → ∃𝑥𝑥, (𝐹𝑋)⟩ ∈ 𝐹))
6 fvex 6424 . . . 4 (𝐹𝑋) ∈ V
7 elimasng 5708 . . . 4 ((𝑋𝑉 ∧ (𝐹𝑋) ∈ V) → ((𝐹𝑋) ∈ (𝐹 “ {𝑋}) ↔ ⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹))
86, 7mpan2 683 . . 3 (𝑋𝑉 → ((𝐹𝑋) ∈ (𝐹 “ {𝑋}) ↔ ⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹))
9 elrn2g 5516 . . . 4 ((𝐹𝑋) ∈ V → ((𝐹𝑋) ∈ ran 𝐹 ↔ ∃𝑥𝑥, (𝐹𝑋)⟩ ∈ 𝐹))
106, 9mp1i 13 . . 3 (𝑋𝑉 → ((𝐹𝑋) ∈ ran 𝐹 ↔ ∃𝑥𝑥, (𝐹𝑋)⟩ ∈ 𝐹))
115, 8, 103imtr4d 286 . 2 (𝑋𝑉 → ((𝐹𝑋) ∈ (𝐹 “ {𝑋}) → (𝐹𝑋) ∈ ran 𝐹))
122, 11syl5bir 235 1 (𝑋𝑉 → ((𝐹𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹𝑋) ∈ ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1653  wex 1875  wcel 2157  Vcvv 3385  {csn 4368  cop 4374  ran crn 5313  cres 5314  cima 5315  cfv 6101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-xp 5318  df-cnv 5320  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fv 6109
This theorem is referenced by:  fvn0fvelrn  6658  funressndmfvrn  41927
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