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Theorem fvrnressn 7155
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 5682 . . 3 (𝐹 “ {𝑋}) = ran (𝐹 ↾ {𝑋})
21eleq2i 2819 . 2 ((𝐹𝑋) ∈ (𝐹 “ {𝑋}) ↔ (𝐹𝑋) ∈ ran (𝐹 ↾ {𝑋}))
3 opeq1 4868 . . . . 5 (𝑥 = 𝑋 → ⟨𝑥, (𝐹𝑋)⟩ = ⟨𝑋, (𝐹𝑋)⟩)
43eleq1d 2812 . . . 4 (𝑥 = 𝑋 → (⟨𝑥, (𝐹𝑋)⟩ ∈ 𝐹 ↔ ⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹))
54spcegv 3581 . . 3 (𝑋𝑉 → (⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹 → ∃𝑥𝑥, (𝐹𝑋)⟩ ∈ 𝐹))
6 fvex 6898 . . . 4 (𝐹𝑋) ∈ V
7 elimasng 6081 . . . 4 ((𝑋𝑉 ∧ (𝐹𝑋) ∈ V) → ((𝐹𝑋) ∈ (𝐹 “ {𝑋}) ↔ ⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹))
86, 7mpan2 688 . . 3 (𝑋𝑉 → ((𝐹𝑋) ∈ (𝐹 “ {𝑋}) ↔ ⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹))
9 elrn2g 5884 . . . 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 242 1 (𝑋𝑉 → ((𝐹𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹𝑋) ∈ ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wex 1773  wcel 2098  Vcvv 3468  {csn 4623  cop 4629  ran crn 5670  cres 5671  cima 5672  cfv 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fv 6545
This theorem is referenced by:  fvn0fvelrnOLD  7157  funressndmfvrn  46326
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