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Theorem rnsnf 45127
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 4648 . . . . . 6 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
21fveq2d 6911 . . . . 5 (𝑥 ∈ {𝐴} → (𝐹𝑥) = (𝐹𝐴))
32mpteq2ia 5251 . . . 4 (𝑥 ∈ {𝐴} ↦ (𝐹𝑥)) = (𝑥 ∈ {𝐴} ↦ (𝐹𝐴))
4 rnsnf.2 . . . . 5 (𝜑𝐹:{𝐴}⟶𝐵)
54feqmptd 6977 . . . 4 (𝜑𝐹 = (𝑥 ∈ {𝐴} ↦ (𝐹𝑥)))
6 rnsnf.1 . . . . 5 (𝜑𝐴𝑉)
7 fvexd 6922 . . . . 5 (𝜑 → (𝐹𝐴) ∈ V)
8 fmptsn 7187 . . . . 5 ((𝐴𝑉 ∧ (𝐹𝐴) ∈ V) → {⟨𝐴, (𝐹𝐴)⟩} = (𝑥 ∈ {𝐴} ↦ (𝐹𝐴)))
96, 7, 8syl2anc 584 . . . 4 (𝜑 → {⟨𝐴, (𝐹𝐴)⟩} = (𝑥 ∈ {𝐴} ↦ (𝐹𝐴)))
103, 5, 93eqtr4a 2801 . . 3 (𝜑𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
1110rneqd 5952 . 2 (𝜑 → ran 𝐹 = ran {⟨𝐴, (𝐹𝐴)⟩})
12 rnsnopg 6243 . . 3 (𝐴𝑉 → ran {⟨𝐴, (𝐹𝐴)⟩} = {(𝐹𝐴)})
136, 12syl 17 . 2 (𝜑 → ran {⟨𝐴, (𝐹𝐴)⟩} = {(𝐹𝐴)})
1411, 13eqtrd 2775 1 (𝜑 → ran 𝐹 = {(𝐹𝐴)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  {csn 4631  cop 4637  cmpt 5231  ran crn 5690  wf 6559  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571
This theorem is referenced by:  fsneqrn  45154  unirnmapsn  45157  sge0sn  46335
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