Theorem fsnunres 6962

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 6455	. . . 4 ⊢ (𝐹 Fn 𝑆 → (𝐹 ↾ 𝑆) = 𝐹)
2	1	adantr 484	. . 3 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → (𝐹 ↾ 𝑆) = 𝐹)
3		ressnop0 6927	. . . 4 ⊢ (¬ 𝑋 ∈ 𝑆 → ({⟨𝑋, 𝑌⟩} ↾ 𝑆) = ∅)
4	3	adantl 485	. . 3 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ({⟨𝑋, 𝑌⟩} ↾ 𝑆) = ∅)
5	2, 4	uneq12d 4054	. 2 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ((𝐹 ↾ 𝑆) ∪ ({⟨𝑋, 𝑌⟩} ↾ 𝑆)) = (𝐹 ∪ ∅))
6		resundir 5840	. 2 ⊢ ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = ((𝐹 ↾ 𝑆) ∪ ({⟨𝑋, 𝑌⟩} ↾ 𝑆))
7		un0 4279	. . 3 ⊢ (𝐹 ∪ ∅) = 𝐹
8	7	eqcomi 2747	. 2 ⊢ 𝐹 = (𝐹 ∪ ∅)
9	5, 6, 8	3eqtr4g 2798	1 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1542   ∈ wcel 2114   ∪ cun 3841  ∅c0 4211  {csn 4516  ⟨cop 4522   ↾ cres 5527   Fn wfn 6334

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 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-xp 5531  df-rel 5532  df-dm 5535  df-res 5537  df-fun 6341  df-fn 6342

This theorem is referenced by:  pgpfaclem1  19324  islindf4  20656