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Theorem rnsnf 38365
Description: The range of a function whose domain is a singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
rnsnf.1 (𝜑𝐴𝑉)
rnsnf.2 (𝜑𝐹:{𝐴}⟶𝐵)
Assertion
Ref Expression
rnsnf (𝜑 → ran 𝐹 = {(𝐹𝐴)})

Proof of Theorem rnsnf
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elsni 4142 . . . . . . 7 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
21fveq2d 6107 . . . . . 6 (𝑥 ∈ {𝐴} → (𝐹𝑥) = (𝐹𝐴))
32mpteq2ia 4668 . . . . 5 (𝑥 ∈ {𝐴} ↦ (𝐹𝑥)) = (𝑥 ∈ {𝐴} ↦ (𝐹𝐴))
43a1i 11 . . . 4 (𝜑 → (𝑥 ∈ {𝐴} ↦ (𝐹𝑥)) = (𝑥 ∈ {𝐴} ↦ (𝐹𝐴)))
5 rnsnf.2 . . . . 5 (𝜑𝐹:{𝐴}⟶𝐵)
65feqmptd 6159 . . . 4 (𝜑𝐹 = (𝑥 ∈ {𝐴} ↦ (𝐹𝑥)))
7 rnsnf.1 . . . . 5 (𝜑𝐴𝑉)
8 fvex 6113 . . . . . 6 (𝐹𝐴) ∈ V
98a1i 11 . . . . 5 (𝜑 → (𝐹𝐴) ∈ V)
10 fmptsn 6338 . . . . 5 ((𝐴𝑉 ∧ (𝐹𝐴) ∈ V) → {⟨𝐴, (𝐹𝐴)⟩} = (𝑥 ∈ {𝐴} ↦ (𝐹𝐴)))
117, 9, 10syl2anc 691 . . . 4 (𝜑 → {⟨𝐴, (𝐹𝐴)⟩} = (𝑥 ∈ {𝐴} ↦ (𝐹𝐴)))
124, 6, 113eqtr4d 2654 . . 3 (𝜑𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
1312rneqd 5274 . 2 (𝜑 → ran 𝐹 = ran {⟨𝐴, (𝐹𝐴)⟩})
14 rnsnopg 5532 . . 3 (𝐴𝑉 → ran {⟨𝐴, (𝐹𝐴)⟩} = {(𝐹𝐴)})
157, 14syl 17 . 2 (𝜑 → ran {⟨𝐴, (𝐹𝐴)⟩} = {(𝐹𝐴)})
1613, 15eqtrd 2644 1 (𝜑 → ran 𝐹 = {(𝐹𝐴)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  Vcvv 3173  {csn 4125  cop 4131  cmpt 4643  ran crn 5039  wf 5800  cfv 5804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812
This theorem is referenced by:  fsneqrn  38398  unirnmapsn  38401  sge0sn  39272
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