Theorem fsnunfv 5716

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 4927	. . . . . . . . 9 ⊢ dom (𝐹 ↾ {𝑋}) = ({𝑋} ∩ dom 𝐹)
2		incom 3327	. . . . . . . . 9 ⊢ ({𝑋} ∩ dom 𝐹) = (dom 𝐹 ∩ {𝑋})
3	1, 2	eqtri 2198	. . . . . . . 8 ⊢ dom (𝐹 ↾ {𝑋}) = (dom 𝐹 ∩ {𝑋})
4		disjsn 3654	. . . . . . . . 9 ⊢ ((dom 𝐹 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ dom 𝐹)
5	4	biimpri 133	. . . . . . . 8 ⊢ (¬ 𝑋 ∈ dom 𝐹 → (dom 𝐹 ∩ {𝑋}) = ∅)
6	3, 5	eqtrid 2222	. . . . . . 7 ⊢ (¬ 𝑋 ∈ dom 𝐹 → dom (𝐹 ↾ {𝑋}) = ∅)
7	6	3ad2ant3 1020	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → dom (𝐹 ↾ {𝑋}) = ∅)
8		relres 4934	. . . . . . 7 ⊢ Rel (𝐹 ↾ {𝑋})
9		reldm0 4844	. . . . . . 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 134	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → (𝐹 ↾ {𝑋}) = ∅)
12		fnsng 5262	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
13	12	3adant3 1017	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
14		fnresdm 5324	. . . . . 6 ⊢ ({⟨𝑋, 𝑌⟩} Fn {𝑋} → ({⟨𝑋, 𝑌⟩} ↾ {𝑋}) = {⟨𝑋, 𝑌⟩})
15	13, 14	syl 14	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ({⟨𝑋, 𝑌⟩} ↾ {𝑋}) = {⟨𝑋, 𝑌⟩})
16	11, 15	uneq12d 3290	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ {𝑋}) ∪ ({⟨𝑋, 𝑌⟩} ↾ {𝑋})) = (∅ ∪ {⟨𝑋, 𝑌⟩}))
17		resundir 4920	. . . 4 ⊢ ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋}) = ((𝐹 ↾ {𝑋}) ∪ ({⟨𝑋, 𝑌⟩} ↾ {𝑋}))
18		uncom 3279	. . . . 5 ⊢ (∅ ∪ {⟨𝑋, 𝑌⟩}) = ({⟨𝑋, 𝑌⟩} ∪ ∅)
19		un0 3456	. . . . 5 ⊢ ({⟨𝑋, 𝑌⟩} ∪ ∅) = {⟨𝑋, 𝑌⟩}
20	18, 19	eqtr2i 2199	. . . 4 ⊢ {⟨𝑋, 𝑌⟩} = (∅ ∪ {⟨𝑋, 𝑌⟩})
21	16, 17, 20	3eqtr4g 2235	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋}) = {⟨𝑋, 𝑌⟩})
22	21	fveq1d 5516	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → (((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋})‘𝑋) = ({⟨𝑋, 𝑌⟩}‘𝑋))
23		snidg 3621	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
24	23	3ad2ant1 1018	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → 𝑋 ∈ {𝑋})
25		fvres 5538	. . 3 ⊢ (𝑋 ∈ {𝑋} → (((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋})‘𝑋) = ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋))
26	24, 25	syl 14	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → (((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋})‘𝑋) = ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋))
27		fvsng 5711	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
28	27	3adant3 1017	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
29	22, 26, 28	3eqtr3d 2218	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋) = 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148   ∪ cun 3127   ∩ cin 3128  ∅c0 3422  {csn 3592  ⟨cop 3595  dom cdm 4625   ↾ cres 4627  Rel wrel 4630   Fn wfn 5210  ‘cfv 5215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-res 4637  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223

This theorem is referenced by:  tfrlemisucaccv  6323  tfr1onlemsucaccv  6339  tfrcllemsucaccv  6352  inftonninf  10436  hashinfom  10751  zfz1isolemiso  10812  fvsetsid  12488