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Theorem rnsnf 45189
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 4643 . . . . . 6 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
21fveq2d 6910 . . . . 5 (𝑥 ∈ {𝐴} → (𝐹𝑥) = (𝐹𝐴))
32mpteq2ia 5245 . . . 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 6921 . . . . 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 2803 . . 3 (𝜑𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
1110rneqd 5949 . 2 (𝜑 → ran 𝐹 = ran {⟨𝐴, (𝐹𝐴)⟩})
12 rnsnopg 6241 . . 3 (𝐴𝑉 → ran {⟨𝐴, (𝐹𝐴)⟩} = {(𝐹𝐴)})
136, 12syl 17 . 2 (𝜑 → ran {⟨𝐴, (𝐹𝐴)⟩} = {(𝐹𝐴)})
1411, 13eqtrd 2777 1 (𝜑 → ran 𝐹 = {(𝐹𝐴)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626  cop 4632  cmpt 5225  ran crn 5686  wf 6557  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569
This theorem is referenced by:  fsneqrn  45216  unirnmapsn  45219  sge0sn  46394
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