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Theorem rnsnf 38830
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 4170 . . . . . . 7 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
21fveq2d 6154 . . . . . 6 (𝑥 ∈ {𝐴} → (𝐹𝑥) = (𝐹𝐴))
32mpteq2ia 4705 . . . . 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 6207 . . . 4 (𝜑𝐹 = (𝑥 ∈ {𝐴} ↦ (𝐹𝑥)))
7 rnsnf.1 . . . . 5 (𝜑𝐴𝑉)
8 fvex 6160 . . . . . 6 (𝐹𝐴) ∈ V
98a1i 11 . . . . 5 (𝜑 → (𝐹𝐴) ∈ V)
10 fmptsn 6388 . . . . 5 ((𝐴𝑉 ∧ (𝐹𝐴) ∈ V) → {⟨𝐴, (𝐹𝐴)⟩} = (𝑥 ∈ {𝐴} ↦ (𝐹𝐴)))
117, 9, 10syl2anc 692 . . . 4 (𝜑 → {⟨𝐴, (𝐹𝐴)⟩} = (𝑥 ∈ {𝐴} ↦ (𝐹𝐴)))
124, 6, 113eqtr4d 2670 . . 3 (𝜑𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
1312rneqd 5317 . 2 (𝜑 → ran 𝐹 = ran {⟨𝐴, (𝐹𝐴)⟩})
14 rnsnopg 5576 . . 3 (𝐴𝑉 → ran {⟨𝐴, (𝐹𝐴)⟩} = {(𝐹𝐴)})
157, 14syl 17 . 2 (𝜑 → ran {⟨𝐴, (𝐹𝐴)⟩} = {(𝐹𝐴)})
1613, 15eqtrd 2660 1 (𝜑 → ran 𝐹 = {(𝐹𝐴)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1992  Vcvv 3191  {csn 4153  cop 4159  cmpt 4678  ran crn 5080  wf 5846  cfv 5850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858
This theorem is referenced by:  fsneqrn  38863  unirnmapsn  38866  sge0sn  39890
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