Theorem rnsnf 41493

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 4584	. . . . . 6 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
2	1	fveq2d 6674	. . . . 5 ⊢ (𝑥 ∈ {𝐴} → (𝐹‘𝑥) = (𝐹‘𝐴))
3	2	mpteq2ia 5157	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↦ (𝐹‘𝑥)) = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝐴))
4		rnsnf.2	. . . . 5 ⊢ (𝜑 → 𝐹:{𝐴}⟶𝐵)
5	4	feqmptd 6733	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝑥)))
6		rnsnf.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7		fvexd 6685	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐴) ∈ V)
8		fmptsn 6929	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ∈ V) → {⟨𝐴, (𝐹‘𝐴)⟩} = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝐴)))
9	6, 7, 8	syl2anc 586	. . . 4 ⊢ (𝜑 → {⟨𝐴, (𝐹‘𝐴)⟩} = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝐴)))
10	3, 5, 9	3eqtr4a 2882	. . 3 ⊢ (𝜑 → 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})
11	10	rneqd 5808	. 2 ⊢ (𝜑 → ran 𝐹 = ran {⟨𝐴, (𝐹‘𝐴)⟩})
12		rnsnopg 6078	. . 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 2856	1 ⊢ (𝜑 → ran 𝐹 = {(𝐹‘𝐴)})

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  Vcvv 3494  {csn 4567  ⟨cop 4573   ↦ cmpt 5146  ran crn 5556  ⟶wf 6351  ‘cfv 6355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363

This theorem is referenced by:  fsneqrn  41523  unirnmapsn  41526  sge0sn  42710