Theorem fsnunres 7189

Description: Recover the original function from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.)

Assertion

Ref	Expression
fsnunres	⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = 𝐹)

Proof of Theorem fsnunres

Step	Hyp	Ref	Expression
1		fnresdm 6655	. . . 4 ⊢ (𝐹 Fn 𝑆 → (𝐹 ↾ 𝑆) = 𝐹)
2	1	adantr 486	. . 3 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → (𝐹 ↾ 𝑆) = 𝐹)
3		ressnop0 7153	. . . 4 ⊢ (¬ 𝑋 ∈ 𝑆 → ({⟨𝑋, 𝑌⟩} ↾ 𝑆) = ∅)
4	3	adantl 487	. . 3 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ({⟨𝑋, 𝑌⟩} ↾ 𝑆) = ∅)
5	2, 4	uneq12d 4119	. 2 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ((𝐹 ↾ 𝑆) ∪ ({⟨𝑋, 𝑌⟩} ↾ 𝑆)) = (𝐹 ∪ ∅))
6		resundir 5991	. 2 ⊢ ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = ((𝐹 ↾ 𝑆) ∪ ({⟨𝑋, 𝑌⟩} ↾ 𝑆))
7		un0 4347	. . 3 ⊢ (𝐹 ∪ ∅) = 𝐹
8	7	eqcomi 2771	. 2 ⊢ 𝐹 = (𝐹 ∪ ∅)
9	5, 6, 8	3eqtr4g 2822	1 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = 𝐹)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∪ cun 3900  ∅c0 4282  {csn 4587  ⟨cop 4593   ↾ cres 5661   Fn wfn 6532

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-dm 5669  df-res 5671  df-fun 6539  df-fn 6540

This theorem is used by:  pgpfaclem1  20211  islindf4  22052  evlextv  34039