Theorem fsnunfv 7207

Description: Recover the added point from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by NM, 18-May-2017.)

Assertion

Ref	Expression
fsnunfv	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋) = 𝑌)

Proof of Theorem fsnunfv

Step	Hyp	Ref	Expression
1		dmres 6032	. . . . . . . . 9 ⊢ dom (𝐹 ↾ {𝑋}) = ({𝑋} ∩ dom 𝐹)
2		incom 4217	. . . . . . . . 9 ⊢ ({𝑋} ∩ dom 𝐹) = (dom 𝐹 ∩ {𝑋})
3	1, 2	eqtri 2763	. . . . . . . 8 ⊢ dom (𝐹 ↾ {𝑋}) = (dom 𝐹 ∩ {𝑋})
4		disjsn 4716	. . . . . . . . 9 ⊢ ((dom 𝐹 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ dom 𝐹)
5	4	biimpri 228	. . . . . . . 8 ⊢ (¬ 𝑋 ∈ dom 𝐹 → (dom 𝐹 ∩ {𝑋}) = ∅)
6	3, 5	eqtrid 2787	. . . . . . 7 ⊢ (¬ 𝑋 ∈ dom 𝐹 → dom (𝐹 ↾ {𝑋}) = ∅)
7	6	3ad2ant3 1134	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → dom (𝐹 ↾ {𝑋}) = ∅)
8		relres 6026	. . . . . . 7 ⊢ Rel (𝐹 ↾ {𝑋})
9		reldm0 5941	. . . . . . 7 ⊢ (Rel (𝐹 ↾ {𝑋}) → ((𝐹 ↾ {𝑋}) = ∅ ↔ dom (𝐹 ↾ {𝑋}) = ∅))
10	8, 9	ax-mp 5	. . . . . 6 ⊢ ((𝐹 ↾ {𝑋}) = ∅ ↔ dom (𝐹 ↾ {𝑋}) = ∅)
11	7, 10	sylibr 234	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → (𝐹 ↾ {𝑋}) = ∅)
12		fnsng 6620	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
13	12	3adant3 1131	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
14		fnresdm 6688	. . . . . 6 ⊢ ({⟨𝑋, 𝑌⟩} Fn {𝑋} → ({⟨𝑋, 𝑌⟩} ↾ {𝑋}) = {⟨𝑋, 𝑌⟩})
15	13, 14	syl 17	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ({⟨𝑋, 𝑌⟩} ↾ {𝑋}) = {⟨𝑋, 𝑌⟩})
16	11, 15	uneq12d 4179	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ {𝑋}) ∪ ({⟨𝑋, 𝑌⟩} ↾ {𝑋})) = (∅ ∪ {⟨𝑋, 𝑌⟩}))
17		resundir 6015	. . . 4 ⊢ ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋}) = ((𝐹 ↾ {𝑋}) ∪ ({⟨𝑋, 𝑌⟩} ↾ {𝑋}))
18		uncom 4168	. . . . 5 ⊢ (∅ ∪ {⟨𝑋, 𝑌⟩}) = ({⟨𝑋, 𝑌⟩} ∪ ∅)
19		un0 4400	. . . . 5 ⊢ ({⟨𝑋, 𝑌⟩} ∪ ∅) = {⟨𝑋, 𝑌⟩}
20	18, 19	eqtr2i 2764	. . . 4 ⊢ {⟨𝑋, 𝑌⟩} = (∅ ∪ {⟨𝑋, 𝑌⟩})
21	16, 17, 20	3eqtr4g 2800	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋}) = {⟨𝑋, 𝑌⟩})
22	21	fveq1d 6909	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → (((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋})‘𝑋) = ({⟨𝑋, 𝑌⟩}‘𝑋))
23		snidg 4665	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
24	23	3ad2ant1 1132	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → 𝑋 ∈ {𝑋})
25	24	fvresd 6927	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → (((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋})‘𝑋) = ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋))
26		fvsng 7200	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
27	26	3adant3 1131	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
28	22, 25, 27	3eqtr3d 2783	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋) = 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ∪ cun 3961   ∩ cin 3962  ∅c0 4339  {csn 4631  ⟨cop 4637  dom cdm 5689   ↾ cres 5691  Rel wrel 5694   Fn wfn 6558  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571

This theorem is referenced by:  f1ounsn  7292  hashf1lem1  14491  cats1un  14756  fvsetsid  17202  islindf4  21876  wlkp1lem3  29708  wlkp1lem7  29712  wlkp1lem8  29713  eupth2eucrct  30246  mapfzcons2  42707  fnchoice  44967  nnsum4primeseven  47725  nnsum4primesevenALTV  47726  isubgr3stgrlem3  47871