Theorem resasplit 6628

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 6535	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
2		fnresdm 6535	. . . 4 ⊢ (𝐺 Fn 𝐵 → (𝐺 ↾ 𝐵) = 𝐺)
3		uneq12 4088	. . . 4 ⊢ (((𝐹 ↾ 𝐴) = 𝐹 ∧ (𝐺 ↾ 𝐵) = 𝐺) → ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)) = (𝐹 ∪ 𝐺))
4	1, 2, 3	syl2an 595	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)) = (𝐹 ∪ 𝐺))
5	4	3adant3 1130	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)) = (𝐹 ∪ 𝐺))
6		inundif 4409	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = 𝐴
7	6	reseq2i 5877	. . . . . . 7 ⊢ (𝐹 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) = (𝐹 ↾ 𝐴)
8		resundi 5894	. . . . . . 7 ⊢ (𝐹 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵)))
9	7, 8	eqtr3i 2768	. . . . . 6 ⊢ (𝐹 ↾ 𝐴) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵)))
10		incom 4131	. . . . . . . . . 10 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
11	10	uneq1i 4089	. . . . . . . . 9 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴)) = ((𝐵 ∩ 𝐴) ∪ (𝐵 ∖ 𝐴))
12		inundif 4409	. . . . . . . . 9 ⊢ ((𝐵 ∩ 𝐴) ∪ (𝐵 ∖ 𝐴)) = 𝐵
13	11, 12	eqtri 2766	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴)) = 𝐵
14	13	reseq2i 5877	. . . . . . 7 ⊢ (𝐺 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴))) = (𝐺 ↾ 𝐵)
15		resundi 5894	. . . . . . 7 ⊢ (𝐺 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴))) = ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))
16	14, 15	eqtr3i 2768	. . . . . 6 ⊢ (𝐺 ↾ 𝐵) = ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))
17	9, 16	uneq12i 4091	. . . . 5 ⊢ ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)) = (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵))) ∪ ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))))
18		simp3 1136	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵)))
19	18	uneq1d 4092	. . . . . 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 4093	. . . . 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 2798	. . . 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 4099	. . . 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 2795	. . 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 4082	. . . 4 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∩ 𝐵))) = (𝐹 ↾ (𝐴 ∩ 𝐵))
25	24	uneq1i 4089	. . 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 2795	. 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 2780	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 1085   = wceq 1539   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882   ↾ cres 5582   Fn wfn 6413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-dm 5590  df-res 5592  df-fun 6420  df-fn 6421

This theorem is referenced by:  fresaun  6629  fresaunres2  6630