Theorem rnsnf 45178

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 4606	. . . . . 6 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
2	1	fveq2d 6862	. . . . 5 ⊢ (𝑥 ∈ {𝐴} → (𝐹‘𝑥) = (𝐹‘𝐴))
3	2	mpteq2ia 5202	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↦ (𝐹‘𝑥)) = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝐴))
4		rnsnf.2	. . . . 5 ⊢ (𝜑 → 𝐹:{𝐴}⟶𝐵)
5	4	feqmptd 6929	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝑥)))
6		rnsnf.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7		fvexd 6873	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐴) ∈ V)
8		fmptsn 7141	. . . . 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 2790	. . 3 ⊢ (𝜑 → 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})
11	10	rneqd 5902	. 2 ⊢ (𝜑 → ran 𝐹 = ran {⟨𝐴, (𝐹‘𝐴)⟩})
12		rnsnopg 6194	. . 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 2764	1 ⊢ (𝜑 → ran 𝐹 = {(𝐹‘𝐴)})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3447  {csn 4589  ⟨cop 4595   ↦ cmpt 5188  ran crn 5639  ⟶wf 6507  ‘cfv 6511

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519

This theorem is referenced by:  fsneqrn  45205  unirnmapsn  45208  sge0sn  46377