Theorem fsnunres 7213

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 6677	. . . 4 ⊢ (𝐹 Fn 𝑆 → (𝐹 ↾ 𝑆) = 𝐹)
2	1	adantr 486	. . 3 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → (𝐹 ↾ 𝑆) = 𝐹)
3		ressnop0 7178	. . . 4 ⊢ (¬ 𝑋 ∈ 𝑆 → ({⟨𝑋, 𝑌⟩} ↾ 𝑆) = ∅)
4	3	adantl 487	. . 3 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ({⟨𝑋, 𝑌⟩} ↾ 𝑆) = ∅)
5	2, 4	uneq12d 4171	. 2 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ((𝐹 ↾ 𝑆) ∪ ({⟨𝑋, 𝑌⟩} ↾ 𝑆)) = (𝐹 ∪ ∅))
6		resundir 5995	. 2 ⊢ ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = ((𝐹 ↾ 𝑆) ∪ ({⟨𝑋, 𝑌⟩} ↾ 𝑆))
7		un0 4347	. . 3 ⊢ (𝐹 ∪ ∅) = 𝐹
8	7	eqcomi 2772	. 2 ⊢ 𝐹 = (𝐹 ∪ ∅)
9	5, 6, 8	3eqtr4g 2823	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 2147   ∪ cun 4152  ∅c0 4281  {csn 4588  ⟨cop 4838   ↾ cres 5657   Fn wfn 6550

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-dm 5665  df-res 5667  df-fun 6557  df-fn 6558

This theorem is used by:  pgpfaclem1  20324  islindf4  22171  evlextv  34241