Theorem fsnunfv 7133

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 5970	. . . . . . . . 9 ⊢ dom (𝐹 ↾ {𝑋}) = ({𝑋} ∩ dom 𝐹)
2		incom 4160	. . . . . . . . 9 ⊢ ({𝑋} ∩ dom 𝐹) = (dom 𝐹 ∩ {𝑋})
3	1, 2	eqtri 2758	. . . . . . . 8 ⊢ dom (𝐹 ↾ {𝑋}) = (dom 𝐹 ∩ {𝑋})
4		disjsn 4667	. . . . . . . . 9 ⊢ ((dom 𝐹 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ dom 𝐹)
5	4	biimpri 228	. . . . . . . 8 ⊢ (¬ 𝑋 ∈ dom 𝐹 → (dom 𝐹 ∩ {𝑋}) = ∅)
6	3, 5	eqtrid 2782	. . . . . . 7 ⊢ (¬ 𝑋 ∈ dom 𝐹 → dom (𝐹 ↾ {𝑋}) = ∅)
7	6	3ad2ant3 1136	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → dom (𝐹 ↾ {𝑋}) = ∅)
8		relres 5963	. . . . . . 7 ⊢ Rel (𝐹 ↾ {𝑋})
9		reldm0 5876	. . . . . . 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 6543	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
13	12	3adant3 1133	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
14		fnresdm 6610	. . . . . 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 4120	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ {𝑋}) ∪ ({⟨𝑋, 𝑌⟩} ↾ {𝑋})) = (∅ ∪ {⟨𝑋, 𝑌⟩}))
17		resundir 5952	. . . 4 ⊢ ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋}) = ((𝐹 ↾ {𝑋}) ∪ ({⟨𝑋, 𝑌⟩} ↾ {𝑋}))
18		uncom 4109	. . . . 5 ⊢ (∅ ∪ {⟨𝑋, 𝑌⟩}) = ({⟨𝑋, 𝑌⟩} ∪ ∅)
19		un0 4345	. . . . 5 ⊢ ({⟨𝑋, 𝑌⟩} ∪ ∅) = {⟨𝑋, 𝑌⟩}
20	18, 19	eqtr2i 2759	. . . 4 ⊢ {⟨𝑋, 𝑌⟩} = (∅ ∪ {⟨𝑋, 𝑌⟩})
21	16, 17, 20	3eqtr4g 2795	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋}) = {⟨𝑋, 𝑌⟩})
22	21	fveq1d 6835	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → (((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋})‘𝑋) = ({⟨𝑋, 𝑌⟩}‘𝑋))
23		snidg 4616	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
24	23	3ad2ant1 1134	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → 𝑋 ∈ {𝑋})
25	24	fvresd 6853	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → (((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋})‘𝑋) = ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋))
26		fvsng 7126	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
27	26	3adant3 1133	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
28	22, 25, 27	3eqtr3d 2778	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋) = 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ∪ cun 3898   ∩ cin 3899  ∅c0 4284  {csn 4579  ⟨cop 4585  dom cdm 5623   ↾ cres 5625  Rel wrel 5628   Fn wfn 6486  ‘cfv 6491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-res 5635  df-iota 6447  df-fun 6493  df-fn 6494  df-fv 6499

This theorem is referenced by:  f1ounsn  7218  hashf1lem1  14380  cats1un  14646  fvsetsid  17097  islindf4  21795  wlkp1lem3  29728  wlkp1lem7  29732  wlkp1lem8  29733  eupth2eucrct  30273  evlextv  33686  mapfzcons2  42998  fnchoice  45311  nnsum4primeseven  48083  nnsum4primesevenALTV  48084  isubgr3stgrlem3  48251