Theorem fresunsn 32907

Description: Recover the original function from a point-added function. See also funresdfunsn 7185 and fsnunres 7184. (Contributed by Thierry Arnoux, 15-Feb-2026.)

Assertion

Ref	Expression
fresunsn	⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, 𝑌⟩}) = 𝐹)

Proof of Theorem fresunsn

Step	Hyp	Ref	Expression
1		resdmdfsn 6029	. . . 4 ⊢ (𝐹 ↾ (V ∖ {𝑋})) = (𝐹 ↾ (dom 𝐹 ∖ {𝑋}))
2		fndm 6636	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3	2	3ad2ant1 1149	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → dom 𝐹 = 𝐴)
4	3	difeq1d 4088	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → (dom 𝐹 ∖ {𝑋}) = (𝐴 ∖ {𝑋}))
5	4	reseq2d 5976	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → (𝐹 ↾ (dom 𝐹 ∖ {𝑋})) = (𝐹 ↾ (𝐴 ∖ {𝑋})))
6	1, 5	eqtr2id 2817	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → (𝐹 ↾ (𝐴 ∖ {𝑋})) = (𝐹 ↾ (V ∖ {𝑋})))
7		simp3 1154	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → (𝐹‘𝑋) = 𝑌)
8	7	eqcomd 2775	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → 𝑌 = (𝐹‘𝑋))
9	8	opeq2d 4846	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → ⟨𝑋, 𝑌⟩ = ⟨𝑋, (𝐹‘𝑋)⟩)
10	9	sneqd 4603	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → {⟨𝑋, 𝑌⟩} = {⟨𝑋, (𝐹‘𝑋)⟩})
11	6, 10	uneq12d 4131	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, 𝑌⟩}) = ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}))
12		fnfun 6633	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
13	12	3ad2ant1 1149	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → Fun 𝐹)
14	2	eleq2d 2855	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ 𝐴))
15	14	biimpar 482	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ dom 𝐹)
16	15	3adant3 1148	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → 𝑋 ∈ dom 𝐹)
17		funresdfunsn 7185	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}) = 𝐹)
18	13, 16, 17	syl2anc 595	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}) = 𝐹)
19	11, 18	eqtrd 2804	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, 𝑌⟩}) = 𝐹)

Syntax hints:   → wi 4   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ∖ cdif 3910   ∪ cun 3911  {csn 4591  ⟨cop 4597  dom cdm 5659   ↾ cres 5661  Fun wfun 6527   Fn wfn 6528  ‘cfv 6533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541

This theorem is referenced by:  evlextv  33873