Theorem rnsnf 39370

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 4194	. . . . . . 7 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
2	1	fveq2d 6195	. . . . . 6 ⊢ (𝑥 ∈ {𝐴} → (𝐹‘𝑥) = (𝐹‘𝐴))
3	2	mpteq2ia 4740	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↦ (𝐹‘𝑥)) = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝐴))
4	3	a1i 11	. . . 4 ⊢ (𝜑 → (𝑥 ∈ {𝐴} ↦ (𝐹‘𝑥)) = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝐴)))
5		rnsnf.2	. . . . 5 ⊢ (𝜑 → 𝐹:{𝐴}⟶𝐵)
6	5	feqmptd 6249	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝑥)))
7		rnsnf.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
8		fvexd 6203	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐴) ∈ V)
9		fmptsn 6433	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ∈ V) → {⟨𝐴, (𝐹‘𝐴)⟩} = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝐴)))
10	7, 8, 9	syl2anc 693	. . . 4 ⊢ (𝜑 → {⟨𝐴, (𝐹‘𝐴)⟩} = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝐴)))
11	4, 6, 10	3eqtr4d 2666	. . 3 ⊢ (𝜑 → 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})
12	11	rneqd 5353	. 2 ⊢ (𝜑 → ran 𝐹 = ran {⟨𝐴, (𝐹‘𝐴)⟩})
13		rnsnopg 5614	. . 3 ⊢ (𝐴 ∈ 𝑉 → ran {⟨𝐴, (𝐹‘𝐴)⟩} = {(𝐹‘𝐴)})
14	7, 13	syl 17	. 2 ⊢ (𝜑 → ran {⟨𝐴, (𝐹‘𝐴)⟩} = {(𝐹‘𝐴)})
15	12, 14	eqtrd 2656	1 ⊢ (𝜑 → ran 𝐹 = {(𝐹‘𝐴)})

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Vcvv 3200  {csn 4177  ⟨cop 4183   ↦ cmpt 4729  ran crn 5115  ⟶wf 5884  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  fsneqrn  39403  unirnmapsn  39406  sge0sn  40596