Theorem rnsnf 45935

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 4608	. . . . . 6 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
2	1	fveq2d 6889	. . . . 5 ⊢ (𝑥 ∈ {𝐴} → (𝐹‘𝑥) = (𝐹‘𝐴))
3	2	mpteq2ia 5208	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↦ (𝐹‘𝑥)) = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝐴))
4		rnsnf.2	. . . . 5 ⊢ (𝜑 → 𝐹:{𝐴}⟶𝐵)
5	4	feqmptd 6953	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝑥)))
6		rnsnf.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7		fvexd 6900	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐴) ∈ V)
8		fmptsn 7169	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ∈ V) → {⟨𝐴, (𝐹‘𝐴)⟩} = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝐴)))
9	6, 7, 8	syl2anc 596	. . . 4 ⊢ (𝜑 → {⟨𝐴, (𝐹‘𝐴)⟩} = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝐴)))
10	3, 5, 9	3eqtr4a 2826	. . 3 ⊢ (𝜑 → 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})
11	10	rneqd 5930	. 2 ⊢ (𝜑 → ran 𝐹 = ran {⟨𝐴, (𝐹‘𝐴)⟩})
12		rnsnopg 6224	. . 3 ⊢ (𝐴 ∈ 𝑉 → ran {⟨𝐴, (𝐹‘𝐴)⟩} = {(𝐹‘𝐴)})
13	6, 12	syl 18	. 2 ⊢ (𝜑 → ran {⟨𝐴, (𝐹‘𝐴)⟩} = {(𝐹‘𝐴)})
14	11, 13	eqtrd 2800	1 ⊢ (𝜑 → ran 𝐹 = {(𝐹‘𝐴)})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  Vcvv 3457  {csn 4591  ⟨cop 4597   ↦ cmpt 5194  ran crn 5664  ⟶wf 6536  ‘cfv 6540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548

This theorem is used by:  fsneqrn  45960  unirnmapsn  45963  sge0sn  47126