Theorem fresaunres2 6707

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 6663	. . . 4 ⊢ (𝐹:𝐴⟶𝐶 → 𝐹 Fn 𝐴)
2		ffn 6663	. . . 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 6705	. . . 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 1161	. . 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 5938	. 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 5954	. . 3 ⊢ (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) ↾ 𝐵) = (((𝐹 ↾ (𝐴 ∩ 𝐵)) ↾ 𝐵) ∪ (((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))) ↾ 𝐵))
8		inss2 4179	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
9		resabs2 5969	. . . . . 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 5954	. . . . 5 ⊢ (((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))) ↾ 𝐵) = (((𝐹 ↾ (𝐴 ∖ 𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵 ∖ 𝐴)) ↾ 𝐵))
12	10, 11	uneq12i 4107	. . . 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 5972	. . . . . . . . 9 ⊢ dom ((𝐹 ↾ (𝐴 ∖ 𝐵)) ↾ 𝐵) = (𝐵 ∩ dom (𝐹 ↾ (𝐴 ∖ 𝐵)))
14		dmres 5972	. . . . . . . . . . 11 ⊢ dom (𝐹 ↾ (𝐴 ∖ 𝐵)) = ((𝐴 ∖ 𝐵) ∩ dom 𝐹)
15	14	ineq2i 4158	. . . . . . . . . 10 ⊢ (𝐵 ∩ dom (𝐹 ↾ (𝐴 ∖ 𝐵))) = (𝐵 ∩ ((𝐴 ∖ 𝐵) ∩ dom 𝐹))
16		disjdif 4413	. . . . . . . . . . . 12 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐵)) = ∅
17	16	ineq1i 4157	. . . . . . . . . . 11 ⊢ ((𝐵 ∩ (𝐴 ∖ 𝐵)) ∩ dom 𝐹) = (∅ ∩ dom 𝐹)
18		inass 4169	. . . . . . . . . . 11 ⊢ ((𝐵 ∩ (𝐴 ∖ 𝐵)) ∩ dom 𝐹) = (𝐵 ∩ ((𝐴 ∖ 𝐵) ∩ dom 𝐹))
19		0in 4338	. . . . . . . . . . 11 ⊢ (∅ ∩ dom 𝐹) = ∅
20	17, 18, 19	3eqtr3i 2768	. . . . . . . . . 10 ⊢ (𝐵 ∩ ((𝐴 ∖ 𝐵) ∩ dom 𝐹)) = ∅
21	15, 20	eqtri 2760	. . . . . . . . 9 ⊢ (𝐵 ∩ dom (𝐹 ↾ (𝐴 ∖ 𝐵))) = ∅
22	13, 21	eqtri 2760	. . . . . . . 8 ⊢ dom ((𝐹 ↾ (𝐴 ∖ 𝐵)) ↾ 𝐵) = ∅
23		relres 5965	. . . . . . . . 9 ⊢ Rel ((𝐹 ↾ (𝐴 ∖ 𝐵)) ↾ 𝐵)
24		reldm0 5878	. . . . . . . . 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 4077	. . . . . . . 8 ⊢ (𝐵 ∖ 𝐴) ⊆ 𝐵
28		resabs2 5969	. . . . . . . 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 4107	. . . . . 6 ⊢ (((𝐹 ↾ (𝐴 ∖ 𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵 ∖ 𝐴)) ↾ 𝐵)) = (∅ ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))
31	30	uneq2i 4106	. . . . 5 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (((𝐹 ↾ (𝐴 ∖ 𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵 ∖ 𝐴)) ↾ 𝐵))) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))))
32		simp3 1139	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵)))
33	32	uneq1d 4108	. . . . . 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 4099	. . . . . . . . . 10 ⊢ (∅ ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))) = ((𝐺 ↾ (𝐵 ∖ 𝐴)) ∪ ∅)
35		un0 4335	. . . . . . . . . 10 ⊢ ((𝐺 ↾ (𝐵 ∖ 𝐴)) ∪ ∅) = (𝐺 ↾ (𝐵 ∖ 𝐴))
36	34, 35	eqtri 2760	. . . . . . . . 9 ⊢ (∅ ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))) = (𝐺 ↾ (𝐵 ∖ 𝐴))
37	36	uneq2i 4106	. . . . . . . 8 ⊢ ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) = ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))
38		resundi 5953	. . . . . . . . 9 ⊢ (𝐺 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴))) = ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))
39		incom 4150	. . . . . . . . . . . . 13 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
40	39	uneq1i 4105	. . . . . . . . . . . 12 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴)) = ((𝐵 ∩ 𝐴) ∪ (𝐵 ∖ 𝐴))
41		inundif 4420	. . . . . . . . . . . 12 ⊢ ((𝐵 ∩ 𝐴) ∪ (𝐵 ∖ 𝐴)) = 𝐵
42	40, 41	eqtri 2760	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴)) = 𝐵
43	42	reseq2i 5936	. . . . . . . . . 10 ⊢ (𝐺 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴))) = (𝐺 ↾ 𝐵)
44		fnresdm 6612	. . . . . . . . . . . 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 2784	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶) → (𝐺 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴))) = 𝐺)
48	38, 47	eqtr3id 2786	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶) → ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))) = 𝐺)
49	37, 48	eqtrid 2784	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶) → ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) = 𝐺)
50	49	3adant3 1133	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) = 𝐺)
51	33, 50	eqtrd 2772	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) = 𝐺)
52	31, 51	eqtrid 2784	. . . 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 2784	. . 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 2784	. 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 2772	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 1542   ∖ cdif 3887   ∪ cun 3888   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  dom cdm 5625   ↾ cres 5627  Rel wrel 5630   Fn wfn 6488  ⟶wf 6489

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-ext 2709  ax-sep 5232  ax-pr 5371

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-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5631  df-rel 5632  df-dm 5635  df-res 5637  df-fun 6495  df-fn 6496  df-f 6497

This theorem is referenced by:  fresaunres1  6708  mapunen  9078  ptuncnv  23785  cvmliftlem10  35495  elmapresaunres2  43220