Theorem fresaunres2 6793

Description: From the union of two functions that agree on the domain overlap, either component can be recovered by restriction. (Contributed by Stefan O'Rear, 9-Oct-2014.)

Assertion

Ref	Expression
fresaunres2	⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺)

Proof of Theorem fresaunres2

Step	Hyp	Ref	Expression
1		ffn 6747	. . . 4 ⊢ (𝐹:𝐴⟶𝐶 → 𝐹 Fn 𝐴)
2		ffn 6747	. . . 4 ⊢ (𝐺:𝐵⟶𝐶 → 𝐺 Fn 𝐵)
3		id 22	. . . 4 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵)) → (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵)))
4		resasplit 6791	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))))
5	1, 2, 3, 4	syl3an 1160	. . 3 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))))
6	5	reseq1d 6008	. 2 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) ↾ 𝐵))
7		resundir 6024	. . 3 ⊢ (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) ↾ 𝐵) = (((𝐹 ↾ (𝐴 ∩ 𝐵)) ↾ 𝐵) ∪ (((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))) ↾ 𝐵))
8		inss2 4259	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
9		resabs2 6038	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐵 → ((𝐹 ↾ (𝐴 ∩ 𝐵)) ↾ 𝐵) = (𝐹 ↾ (𝐴 ∩ 𝐵)))
10	8, 9	ax-mp 5	. . . . 5 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)) ↾ 𝐵) = (𝐹 ↾ (𝐴 ∩ 𝐵))
11		resundir 6024	. . . . 5 ⊢ (((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))) ↾ 𝐵) = (((𝐹 ↾ (𝐴 ∖ 𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵 ∖ 𝐴)) ↾ 𝐵))
12	10, 11	uneq12i 4189	. . . 4 ⊢ (((𝐹 ↾ (𝐴 ∩ 𝐵)) ↾ 𝐵) ∪ (((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))) ↾ 𝐵)) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (((𝐹 ↾ (𝐴 ∖ 𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵 ∖ 𝐴)) ↾ 𝐵)))
13		dmres 6041	. . . . . . . . 9 ⊢ dom ((𝐹 ↾ (𝐴 ∖ 𝐵)) ↾ 𝐵) = (𝐵 ∩ dom (𝐹 ↾ (𝐴 ∖ 𝐵)))
14		dmres 6041	. . . . . . . . . . 11 ⊢ dom (𝐹 ↾ (𝐴 ∖ 𝐵)) = ((𝐴 ∖ 𝐵) ∩ dom 𝐹)
15	14	ineq2i 4238	. . . . . . . . . 10 ⊢ (𝐵 ∩ dom (𝐹 ↾ (𝐴 ∖ 𝐵))) = (𝐵 ∩ ((𝐴 ∖ 𝐵) ∩ dom 𝐹))
16		disjdif 4495	. . . . . . . . . . . 12 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐵)) = ∅
17	16	ineq1i 4237	. . . . . . . . . . 11 ⊢ ((𝐵 ∩ (𝐴 ∖ 𝐵)) ∩ dom 𝐹) = (∅ ∩ dom 𝐹)
18		inass 4249	. . . . . . . . . . 11 ⊢ ((𝐵 ∩ (𝐴 ∖ 𝐵)) ∩ dom 𝐹) = (𝐵 ∩ ((𝐴 ∖ 𝐵) ∩ dom 𝐹))
19		0in 4420	. . . . . . . . . . 11 ⊢ (∅ ∩ dom 𝐹) = ∅
20	17, 18, 19	3eqtr3i 2776	. . . . . . . . . 10 ⊢ (𝐵 ∩ ((𝐴 ∖ 𝐵) ∩ dom 𝐹)) = ∅
21	15, 20	eqtri 2768	. . . . . . . . 9 ⊢ (𝐵 ∩ dom (𝐹 ↾ (𝐴 ∖ 𝐵))) = ∅
22	13, 21	eqtri 2768	. . . . . . . 8 ⊢ dom ((𝐹 ↾ (𝐴 ∖ 𝐵)) ↾ 𝐵) = ∅
23		relres 6035	. . . . . . . . 9 ⊢ Rel ((𝐹 ↾ (𝐴 ∖ 𝐵)) ↾ 𝐵)
24		reldm0 5952	. . . . . . . . 9 ⊢ (Rel ((𝐹 ↾ (𝐴 ∖ 𝐵)) ↾ 𝐵) → (((𝐹 ↾ (𝐴 ∖ 𝐵)) ↾ 𝐵) = ∅ ↔ dom ((𝐹 ↾ (𝐴 ∖ 𝐵)) ↾ 𝐵) = ∅))
25	23, 24	ax-mp 5	. . . . . . . 8 ⊢ (((𝐹 ↾ (𝐴 ∖ 𝐵)) ↾ 𝐵) = ∅ ↔ dom ((𝐹 ↾ (𝐴 ∖ 𝐵)) ↾ 𝐵) = ∅)
26	22, 25	mpbir 231	. . . . . . 7 ⊢ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ↾ 𝐵) = ∅
27		difss 4159	. . . . . . . 8 ⊢ (𝐵 ∖ 𝐴) ⊆ 𝐵
28		resabs2 6038	. . . . . . . 8 ⊢ ((𝐵 ∖ 𝐴) ⊆ 𝐵 → ((𝐺 ↾ (𝐵 ∖ 𝐴)) ↾ 𝐵) = (𝐺 ↾ (𝐵 ∖ 𝐴)))
29	27, 28	ax-mp 5	. . . . . . 7 ⊢ ((𝐺 ↾ (𝐵 ∖ 𝐴)) ↾ 𝐵) = (𝐺 ↾ (𝐵 ∖ 𝐴))
30	26, 29	uneq12i 4189	. . . . . 6 ⊢ (((𝐹 ↾ (𝐴 ∖ 𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵 ∖ 𝐴)) ↾ 𝐵)) = (∅ ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))
31	30	uneq2i 4188	. . . . 5 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (((𝐹 ↾ (𝐴 ∖ 𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵 ∖ 𝐴)) ↾ 𝐵))) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))))
32		simp3 1138	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵)))
33	32	uneq1d 4190	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) = ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))))
34		uncom 4181	. . . . . . . . . 10 ⊢ (∅ ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))) = ((𝐺 ↾ (𝐵 ∖ 𝐴)) ∪ ∅)
35		un0 4417	. . . . . . . . . 10 ⊢ ((𝐺 ↾ (𝐵 ∖ 𝐴)) ∪ ∅) = (𝐺 ↾ (𝐵 ∖ 𝐴))
36	34, 35	eqtri 2768	. . . . . . . . 9 ⊢ (∅ ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))) = (𝐺 ↾ (𝐵 ∖ 𝐴))
37	36	uneq2i 4188	. . . . . . . 8 ⊢ ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) = ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))
38		resundi 6023	. . . . . . . . 9 ⊢ (𝐺 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴))) = ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))
39		incom 4230	. . . . . . . . . . . . 13 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
40	39	uneq1i 4187	. . . . . . . . . . . 12 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴)) = ((𝐵 ∩ 𝐴) ∪ (𝐵 ∖ 𝐴))
41		inundif 4502	. . . . . . . . . . . 12 ⊢ ((𝐵 ∩ 𝐴) ∪ (𝐵 ∖ 𝐴)) = 𝐵
42	40, 41	eqtri 2768	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴)) = 𝐵
43	42	reseq2i 6006	. . . . . . . . . 10 ⊢ (𝐺 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴))) = (𝐺 ↾ 𝐵)
44		fnresdm 6699	. . . . . . . . . . . 12 ⊢ (𝐺 Fn 𝐵 → (𝐺 ↾ 𝐵) = 𝐺)
45	2, 44	syl 17	. . . . . . . . . . 11 ⊢ (𝐺:𝐵⟶𝐶 → (𝐺 ↾ 𝐵) = 𝐺)
46	45	adantl 481	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶) → (𝐺 ↾ 𝐵) = 𝐺)
47	43, 46	eqtrid 2792	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶) → (𝐺 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴))) = 𝐺)
48	38, 47	eqtr3id 2794	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶) → ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))) = 𝐺)
49	37, 48	eqtrid 2792	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶) → ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) = 𝐺)
50	49	3adant3 1132	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) = 𝐺)
51	33, 50	eqtrd 2780	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) = 𝐺)
52	31, 51	eqtrid 2792	. . . 4 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (((𝐹 ↾ (𝐴 ∖ 𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵 ∖ 𝐴)) ↾ 𝐵))) = 𝐺)
53	12, 52	eqtrid 2792	. . 3 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (((𝐹 ↾ (𝐴 ∩ 𝐵)) ↾ 𝐵) ∪ (((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))) ↾ 𝐵)) = 𝐺)
54	7, 53	eqtrid 2792	. 2 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) ↾ 𝐵) = 𝐺)
55	6, 54	eqtrd 2780	1 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∖ cdif 3973   ∪ cun 3974   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  dom cdm 5700   ↾ cres 5702  Rel wrel 5705   Fn wfn 6568  ⟶wf 6569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-dm 5710  df-res 5712  df-fun 6575  df-fn 6576  df-f 6577

This theorem is referenced by:  fresaunres1  6794  mapunen  9212  ptuncnv  23836  cvmliftlem10  35262  elmapresaunres2  42727