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Theorem rnsnf 39684
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 4227 . . . . . . 7 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
21fveq2d 6233 . . . . . 6 (𝑥 ∈ {𝐴} → (𝐹𝑥) = (𝐹𝐴))
32mpteq2ia 4773 . . . . 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 6288 . . . 4 (𝜑𝐹 = (𝑥 ∈ {𝐴} ↦ (𝐹𝑥)))
7 rnsnf.1 . . . . 5 (𝜑𝐴𝑉)
8 fvexd 6241 . . . . 5 (𝜑 → (𝐹𝐴) ∈ V)
9 fmptsn 6474 . . . . 5 ((𝐴𝑉 ∧ (𝐹𝐴) ∈ V) → {⟨𝐴, (𝐹𝐴)⟩} = (𝑥 ∈ {𝐴} ↦ (𝐹𝐴)))
107, 8, 9syl2anc 694 . . . 4 (𝜑 → {⟨𝐴, (𝐹𝐴)⟩} = (𝑥 ∈ {𝐴} ↦ (𝐹𝐴)))
114, 6, 103eqtr4d 2695 . . 3 (𝜑𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
1211rneqd 5385 . 2 (𝜑 → ran 𝐹 = ran {⟨𝐴, (𝐹𝐴)⟩})
13 rnsnopg 5650 . . 3 (𝐴𝑉 → ran {⟨𝐴, (𝐹𝐴)⟩} = {(𝐹𝐴)})
147, 13syl 17 . 2 (𝜑 → ran {⟨𝐴, (𝐹𝐴)⟩} = {(𝐹𝐴)})
1512, 14eqtrd 2685 1 (𝜑 → ran 𝐹 = {(𝐹𝐴)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  {csn 4210  cop 4216  cmpt 4762  ran crn 5144  wf 5922  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934
This theorem is referenced by:  fsneqrn  39717  unirnmapsn  39720  sge0sn  40914
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