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	⊢ (𝜑 → 𝐴 ∈ 𝑉)
rnsnf.2	⊢ (𝜑 → 𝐹:{𝐴}⟶𝐵)

Assertion

Ref	Expression
rnsnf	⊢ (𝜑 → ran 𝐹 = {(𝐹‘𝐴)})

Proof of Theorem rnsnf

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elsni 4643	. . . . . 6 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
2	1	fveq2d 6910	. . . . 5 ⊢ (𝑥 ∈ {𝐴} → (𝐹‘𝑥) = (𝐹‘𝐴))
3	2	mpteq2ia 5245	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↦ (𝐹‘𝑥)) = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝐴))
4		rnsnf.2	. . . . 5 ⊢ (𝜑 → 𝐹:{𝐴}⟶𝐵)
5	4	feqmptd 6977	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝑥)))
6		rnsnf.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7		fvexd 6921	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐴) ∈ V)
8		fmptsn 7187	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ∈ V) → {⟨𝐴, (𝐹‘𝐴)⟩} = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝐴)))
9	6, 7, 8	syl2anc 584	. . . 4 ⊢ (𝜑 → {⟨𝐴, (𝐹‘𝐴)⟩} = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝐴)))
10	3, 5, 9	3eqtr4a 2803	. . 3 ⊢ (𝜑 → 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})
11	10	rneqd 5949	. 2 ⊢ (𝜑 → ran 𝐹 = ran {⟨𝐴, (𝐹‘𝐴)⟩})
12		rnsnopg 6241	. . 3 ⊢ (𝐴 ∈ 𝑉 → ran {⟨𝐴, (𝐹‘𝐴)⟩} = {(𝐹‘𝐴)})
13	6, 12	syl 17	. 2 ⊢ (𝜑 → ran {⟨𝐴, (𝐹‘𝐴)⟩} = {(𝐹‘𝐴)})
14	11, 13	eqtrd 2777	1 ⊢ (𝜑 → ran 𝐹 = {(𝐹‘𝐴)})

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