Theorem rnressnsn 32662

Description: The range of a restriction to a singleton is a singleton. See dmressnsn 5976. (Contributed by Thierry Arnoux, 25-Jan-2026.)

Assertion

Ref	Expression
rnressnsn	⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ran (𝐹 ↾ {𝐴}) = {(𝐹‘𝐴)})

Proof of Theorem rnressnsn

Step	Hyp	Ref	Expression
1		funfn 6516	. . . 4 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		fnressn 7097	. . . 4 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹 ↾ {𝐴}) = {⟨𝐴, (𝐹‘𝐴)⟩})
3	1, 2	sylanb 581	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹 ↾ {𝐴}) = {⟨𝐴, (𝐹‘𝐴)⟩})
4	3	rneqd 5882	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ran (𝐹 ↾ {𝐴}) = ran {⟨𝐴, (𝐹‘𝐴)⟩})
5		rnsnopg 6173	. . 3 ⊢ (𝐴 ∈ dom 𝐹 → ran {⟨𝐴, (𝐹‘𝐴)⟩} = {(𝐹‘𝐴)})
6	5	adantl 481	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ran {⟨𝐴, (𝐹‘𝐴)⟩} = {(𝐹‘𝐴)})
7	4, 6	eqtrd 2768	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ran (𝐹 ↾ {𝐴}) = {(𝐹‘𝐴)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {csn 4575  ⟨cop 4581  dom cdm 5619  ran crn 5620   ↾ cres 5621  Fun wfun 6480   Fn wfn 6481  ‘cfv 6486

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494

This theorem is referenced by:  esplyind  33613