Theorem fresunsn 32724

Description: Recover the original function from a point-added function. See also funresdfunsn 7140 and fsnunres 7139. (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		fnrel 6594	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2	1	3ad2ant1 1139	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → Rel 𝐹)
3		resdmdfsn 5990	. . . . 5 ⊢ (Rel 𝐹 → (𝐹 ↾ (V ∖ {𝑋})) = (𝐹 ↾ (dom 𝐹 ∖ {𝑋})))
4	2, 3	syl 17	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → (𝐹 ↾ (V ∖ {𝑋})) = (𝐹 ↾ (dom 𝐹 ∖ {𝑋})))
5		fndm 6595	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6	5	3ad2ant1 1139	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → dom 𝐹 = 𝐴)
7	6	difeq1d 4063	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → (dom 𝐹 ∖ {𝑋}) = (𝐴 ∖ {𝑋}))
8	7	reseq2d 5938	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → (𝐹 ↾ (dom 𝐹 ∖ {𝑋})) = (𝐹 ↾ (𝐴 ∖ {𝑋})))
9	4, 8	eqtr2d 2776	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → (𝐹 ↾ (𝐴 ∖ {𝑋})) = (𝐹 ↾ (V ∖ {𝑋})))
10		simp3 1144	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → (𝐹‘𝑋) = 𝑌)
11	10	eqcomd 2746	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → 𝑌 = (𝐹‘𝑋))
12	11	opeq2d 4818	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → ⟨𝑋, 𝑌⟩ = ⟨𝑋, (𝐹‘𝑋)⟩)
13	12	sneqd 4574	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → {⟨𝑋, 𝑌⟩} = {⟨𝑋, (𝐹‘𝑋)⟩})
14	9, 13	uneq12d 4106	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, 𝑌⟩}) = ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}))
15		fnfun 6592	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
16	15	3ad2ant1 1139	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → Fun 𝐹)
17	5	eleq2d 2826	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ 𝐴))
18	17	biimpar 478	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ dom 𝐹)
19	18	3adant3 1138	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → 𝑋 ∈ dom 𝐹)
20		funresdfunsn 7140	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}) = 𝐹)
21	16, 19, 20	syl2anc 590	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}) = 𝐹)
22	14, 21	eqtrd 2775	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, 𝑌⟩}) = 𝐹)

Syntax hints:   → wi 4   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  Vcvv 3432   ∖ cdif 3887   ∪ cun 3888  {csn 4562  ⟨cop 4568  dom cdm 5625   ↾ cres 5627  Rel wrel 5630  Fun wfun 6486   Fn wfn 6487  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500

This theorem is referenced by:  evlextv  33733