Theorem resasplit 6704

Description: If two functions agree on their common domain, express their union as a union of three functions with pairwise disjoint domains. (Contributed by Stefan O'Rear, 9-Oct-2014.)

Assertion

Ref	Expression
resasplit	⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))))

Proof of Theorem resasplit

Step	Hyp	Ref	Expression
1		fnresdm 6611	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
2		fnresdm 6611	. . . 4 ⊢ (𝐺 Fn 𝐵 → (𝐺 ↾ 𝐵) = 𝐺)
3		uneq12 4115	. . . 4 ⊢ (((𝐹 ↾ 𝐴) = 𝐹 ∧ (𝐺 ↾ 𝐵) = 𝐺) → ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)) = (𝐹 ∪ 𝐺))
4	1, 2, 3	syl2an 596	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)) = (𝐹 ∪ 𝐺))
5	4	3adant3 1132	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)) = (𝐹 ∪ 𝐺))
6		inundif 4431	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = 𝐴
7	6	reseq2i 5935	. . . . . . 7 ⊢ (𝐹 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) = (𝐹 ↾ 𝐴)
8		resundi 5952	. . . . . . 7 ⊢ (𝐹 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵)))
9	7, 8	eqtr3i 2761	. . . . . 6 ⊢ (𝐹 ↾ 𝐴) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵)))
10		incom 4161	. . . . . . . . . 10 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
11	10	uneq1i 4116	. . . . . . . . 9 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴)) = ((𝐵 ∩ 𝐴) ∪ (𝐵 ∖ 𝐴))
12		inundif 4431	. . . . . . . . 9 ⊢ ((𝐵 ∩ 𝐴) ∪ (𝐵 ∖ 𝐴)) = 𝐵
13	11, 12	eqtri 2759	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴)) = 𝐵
14	13	reseq2i 5935	. . . . . . 7 ⊢ (𝐺 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴))) = (𝐺 ↾ 𝐵)
15		resundi 5952	. . . . . . 7 ⊢ (𝐺 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴))) = ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))
16	14, 15	eqtr3i 2761	. . . . . 6 ⊢ (𝐺 ↾ 𝐵) = ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))
17	9, 16	uneq12i 4118	. . . . 5 ⊢ ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)) = (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵))) ∪ ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))))
18		simp3 1138	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵)))
19	18	uneq1d 4119	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))) = ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))))
20	19	uneq2d 4120	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵))) ∪ ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) = (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵))) ∪ ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))))
21	17, 20	eqtr4id 2790	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)) = (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵))) ∪ ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))))
22		un4 4127	. . . 4 ⊢ (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵))) ∪ ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) = (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∩ 𝐵))) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))))
23	21, 22	eqtrdi 2787	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)) = (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∩ 𝐵))) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))))
24		unidm 4109	. . . 4 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∩ 𝐵))) = (𝐹 ↾ (𝐴 ∩ 𝐵))
25	24	uneq1i 4116	. . 3 ⊢ (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∩ 𝐵))) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))))
26	23, 25	eqtrdi 2787	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))))
27	5, 26	eqtr3d 2773	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∖ cdif 3898   ∪ cun 3899   ∩ cin 3900   ↾ cres 5626   Fn wfn 6487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-dm 5634  df-res 5636  df-fun 6494  df-fn 6495

This theorem is referenced by:  fresaun  6705  fresaunres2  6706