Theorem fresaun 6731

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 1136	. . . 4 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → 𝐹:𝐴⟶𝐶)
2		inss1 4200	. . . 4 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
3		fssres 6726	. . . 4 ⊢ ((𝐹:𝐴⟶𝐶 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)⟶𝐶)
4	1, 2, 3	sylancl 586	. . 3 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)⟶𝐶)
5		difss 4099	. . . . 5 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
6		fssres 6726	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐶 ∧ (𝐴 ∖ 𝐵) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)⟶𝐶)
7	1, 5, 6	sylancl 586	. . . 4 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)⟶𝐶)
8		simp2 1137	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → 𝐺:𝐵⟶𝐶)
9		difss 4099	. . . . 5 ⊢ (𝐵 ∖ 𝐴) ⊆ 𝐵
10		fssres 6726	. . . . 5 ⊢ ((𝐺:𝐵⟶𝐶 ∧ (𝐵 ∖ 𝐴) ⊆ 𝐵) → (𝐺 ↾ (𝐵 ∖ 𝐴)):(𝐵 ∖ 𝐴)⟶𝐶)
11	8, 9, 10	sylancl 586	. . . 4 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐺 ↾ (𝐵 ∖ 𝐴)):(𝐵 ∖ 𝐴)⟶𝐶)
12		indifdir 4258	. . . . . 6 ⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ((𝐴 ∩ (𝐵 ∖ 𝐴)) ∖ (𝐵 ∩ (𝐵 ∖ 𝐴)))
13		disjdif 4435	. . . . . . 7 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
14	13	difeq1i 4085	. . . . . 6 ⊢ ((𝐴 ∩ (𝐵 ∖ 𝐴)) ∖ (𝐵 ∩ (𝐵 ∖ 𝐴))) = (∅ ∖ (𝐵 ∩ (𝐵 ∖ 𝐴)))
15		0dif 4368	. . . . . 6 ⊢ (∅ ∖ (𝐵 ∩ (𝐵 ∖ 𝐴))) = ∅
16	12, 14, 15	3eqtri 2756	. . . . 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 6724	. . 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 4247	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ∩ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))) = (((𝐴 ∩ 𝐵) ∩ (𝐴 ∖ 𝐵)) ∪ ((𝐴 ∩ 𝐵) ∩ (𝐵 ∖ 𝐴)))
20		inass 4191	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∩ (𝐵 ∩ (𝐴 ∖ 𝐵)))
21		disjdif 4435	. . . . . . . 8 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐵)) = ∅
22	21	ineq2i 4180	. . . . . . 7 ⊢ (𝐴 ∩ (𝐵 ∩ (𝐴 ∖ 𝐵))) = (𝐴 ∩ ∅)
23		in0 4358	. . . . . . 7 ⊢ (𝐴 ∩ ∅) = ∅
24	20, 22, 23	3eqtri 2756	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐴 ∖ 𝐵)) = ∅
25		incom 4172	. . . . . . . 8 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
26	25	ineq1i 4179	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ((𝐵 ∩ 𝐴) ∩ (𝐵 ∖ 𝐴))
27		inass 4191	. . . . . . . 8 ⊢ ((𝐵 ∩ 𝐴) ∩ (𝐵 ∖ 𝐴)) = (𝐵 ∩ (𝐴 ∩ (𝐵 ∖ 𝐴)))
28	13	ineq2i 4180	. . . . . . . 8 ⊢ (𝐵 ∩ (𝐴 ∩ (𝐵 ∖ 𝐴))) = (𝐵 ∩ ∅)
29		in0 4358	. . . . . . . 8 ⊢ (𝐵 ∩ ∅) = ∅
30	27, 28, 29	3eqtri 2756	. . . . . . 7 ⊢ ((𝐵 ∩ 𝐴) ∩ (𝐵 ∖ 𝐴)) = ∅
31	26, 30	eqtri 2752	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ∅
32	24, 31	uneq12i 4129	. . . . 5 ⊢ (((𝐴 ∩ 𝐵) ∩ (𝐴 ∖ 𝐵)) ∪ ((𝐴 ∩ 𝐵) ∩ (𝐵 ∖ 𝐴))) = (∅ ∪ ∅)
33		un0 4357	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
34	19, 32, 33	3eqtri 2756	. . . 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 6724	. 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 4136	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ∪ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))) = ((𝐴 ∖ 𝐵) ∪ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴)))
38	25	uneq1i 4127	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴)) = ((𝐵 ∩ 𝐴) ∪ (𝐵 ∖ 𝐴))
39		inundif 4442	. . . . . . 7 ⊢ ((𝐵 ∩ 𝐴) ∪ (𝐵 ∖ 𝐴)) = 𝐵
40	38, 39	eqtri 2752	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴)) = 𝐵
41	40	uneq2i 4128	. . . . 5 ⊢ ((𝐴 ∖ 𝐵) ∪ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴))) = ((𝐴 ∖ 𝐵) ∪ 𝐵)
42		undif1 4439	. . . . 5 ⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐴 ∪ 𝐵)
43	37, 41, 42	3eqtri 2756	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ∪ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))) = (𝐴 ∪ 𝐵)
44	43	feq2i 6680	. . 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 6688	. . . . 5 ⊢ (𝐹:𝐴⟶𝐶 → 𝐹 Fn 𝐴)
46		ffn 6688	. . . . 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 6730	. . . . 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 1160	. . . 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 6670	. . 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 1086   = wceq 1540   ∖ cdif 3911   ∪ cun 3912   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296   ↾ cres 5640   Fn wfn 6506  ⟶wf 6507

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-fun 6513  df-fn 6514  df-f 6515

This theorem is referenced by:  elmapresaun  8853  cvmliftlem10  35281