Theorem fresaun 6711

Description: The union of two functions which agree on their common domain is a function. (Contributed by Stefan O'Rear, 9-Oct-2014.)

Assertion

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

Proof of Theorem fresaun

Step	Hyp	Ref	Expression
1		simp1 1137	. . . 4 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → 𝐹:𝐴⟶𝐶)
2		inss1 4177	. . . 4 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
3		fssres 6706	. . . 4 ⊢ ((𝐹:𝐴⟶𝐶 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)⟶𝐶)
4	1, 2, 3	sylancl 587	. . 3 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)⟶𝐶)
5		difss 4076	. . . . 5 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
6		fssres 6706	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐶 ∧ (𝐴 ∖ 𝐵) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)⟶𝐶)
7	1, 5, 6	sylancl 587	. . . 4 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)⟶𝐶)
8		simp2 1138	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → 𝐺:𝐵⟶𝐶)
9		difss 4076	. . . . 5 ⊢ (𝐵 ∖ 𝐴) ⊆ 𝐵
10		fssres 6706	. . . . 5 ⊢ ((𝐺:𝐵⟶𝐶 ∧ (𝐵 ∖ 𝐴) ⊆ 𝐵) → (𝐺 ↾ (𝐵 ∖ 𝐴)):(𝐵 ∖ 𝐴)⟶𝐶)
11	8, 9, 10	sylancl 587	. . . 4 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐺 ↾ (𝐵 ∖ 𝐴)):(𝐵 ∖ 𝐴)⟶𝐶)
12		indifdir 4235	. . . . . 6 ⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ((𝐴 ∩ (𝐵 ∖ 𝐴)) ∖ (𝐵 ∩ (𝐵 ∖ 𝐴)))
13		disjdif 4412	. . . . . . 7 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
14	13	difeq1i 4062	. . . . . 6 ⊢ ((𝐴 ∩ (𝐵 ∖ 𝐴)) ∖ (𝐵 ∩ (𝐵 ∖ 𝐴))) = (∅ ∖ (𝐵 ∩ (𝐵 ∖ 𝐴)))
15		0dif 4345	. . . . . 6 ⊢ (∅ ∖ (𝐵 ∩ (𝐵 ∖ 𝐴))) = ∅
16	12, 14, 15	3eqtri 2763	. . . . 5 ⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ∅
17	16	a1i 11	. . . 4 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ∅)
18	7, 11, 17	fun2d 6704	. . 3 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))):((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))⟶𝐶)
19		indi 4224	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ∩ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))) = (((𝐴 ∩ 𝐵) ∩ (𝐴 ∖ 𝐵)) ∪ ((𝐴 ∩ 𝐵) ∩ (𝐵 ∖ 𝐴)))
20		inass 4168	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∩ (𝐵 ∩ (𝐴 ∖ 𝐵)))
21		disjdif 4412	. . . . . . . 8 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐵)) = ∅
22	21	ineq2i 4157	. . . . . . 7 ⊢ (𝐴 ∩ (𝐵 ∩ (𝐴 ∖ 𝐵))) = (𝐴 ∩ ∅)
23		in0 4335	. . . . . . 7 ⊢ (𝐴 ∩ ∅) = ∅
24	20, 22, 23	3eqtri 2763	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐴 ∖ 𝐵)) = ∅
25		incom 4149	. . . . . . . 8 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
26	25	ineq1i 4156	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ((𝐵 ∩ 𝐴) ∩ (𝐵 ∖ 𝐴))
27		inass 4168	. . . . . . . 8 ⊢ ((𝐵 ∩ 𝐴) ∩ (𝐵 ∖ 𝐴)) = (𝐵 ∩ (𝐴 ∩ (𝐵 ∖ 𝐴)))
28	13	ineq2i 4157	. . . . . . . 8 ⊢ (𝐵 ∩ (𝐴 ∩ (𝐵 ∖ 𝐴))) = (𝐵 ∩ ∅)
29		in0 4335	. . . . . . . 8 ⊢ (𝐵 ∩ ∅) = ∅
30	27, 28, 29	3eqtri 2763	. . . . . . 7 ⊢ ((𝐵 ∩ 𝐴) ∩ (𝐵 ∖ 𝐴)) = ∅
31	26, 30	eqtri 2759	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ∅
32	24, 31	uneq12i 4106	. . . . 5 ⊢ (((𝐴 ∩ 𝐵) ∩ (𝐴 ∖ 𝐵)) ∪ ((𝐴 ∩ 𝐵) ∩ (𝐵 ∖ 𝐴))) = (∅ ∪ ∅)
33		un0 4334	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
34	19, 32, 33	3eqtri 2763	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ∩ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))) = ∅
35	34	a1i 11	. . 3 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐴 ∩ 𝐵) ∩ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))) = ∅)
36	4, 18, 35	fun2d 6704	. 2 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))):((𝐴 ∩ 𝐵) ∪ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)))⟶𝐶)
37		un12 4113	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ∪ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))) = ((𝐴 ∖ 𝐵) ∪ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴)))
38	25	uneq1i 4104	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴)) = ((𝐵 ∩ 𝐴) ∪ (𝐵 ∖ 𝐴))
39		inundif 4419	. . . . . . 7 ⊢ ((𝐵 ∩ 𝐴) ∪ (𝐵 ∖ 𝐴)) = 𝐵
40	38, 39	eqtri 2759	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴)) = 𝐵
41	40	uneq2i 4105	. . . . 5 ⊢ ((𝐴 ∖ 𝐵) ∪ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴))) = ((𝐴 ∖ 𝐵) ∪ 𝐵)
42		undif1 4416	. . . . 5 ⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐴 ∪ 𝐵)
43	37, 41, 42	3eqtri 2763	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ∪ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))) = (𝐴 ∪ 𝐵)
44	43	feq2i 6660	. . 3 ⊢ (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))):((𝐴 ∩ 𝐵) ∪ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)))⟶𝐶 ↔ ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))):(𝐴 ∪ 𝐵)⟶𝐶)
45		ffn 6668	. . . . 5 ⊢ (𝐹:𝐴⟶𝐶 → 𝐹 Fn 𝐴)
46		ffn 6668	. . . . 5 ⊢ (𝐺:𝐵⟶𝐶 → 𝐺 Fn 𝐵)
47		id 22	. . . . 5 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵)) → (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵)))
48		resasplit 6710	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))))
49	45, 46, 47, 48	syl3an 1161	. . . 4 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))))
50	49	feq1d 6650	. . 3 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶 ↔ ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))):(𝐴 ∪ 𝐵)⟶𝐶))
51	44, 50	bitr4id 290	. 2 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))):((𝐴 ∩ 𝐵) ∪ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)))⟶𝐶 ↔ (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶))
52	36, 51	mpbid 232	1 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∖ cdif 3886   ∪ cun 3887   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273   ↾ cres 5633   Fn wfn 6493  ⟶wf 6494

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-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

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 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-fun 6500  df-fn 6501  df-f 6502

This theorem is referenced by:  elmapresaun  8828  cvmliftlem10  35476