Theorem resasplitss 5437

Description: If two functions agree on their common domain, their union contains a union of three functions with pairwise disjoint domains. If we assumed the law of the excluded middle, this would be equality rather than subset. (Contributed by Jim Kingdon, 28-Dec-2018.)

Assertion

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

Proof of Theorem resasplitss

Step	Hyp	Ref	Expression
1		unidm 3306	. . . 4 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∩ 𝐵))) = (𝐹 ↾ (𝐴 ∩ 𝐵))
2	1	uneq1i 3313	. . 3 ⊢ (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∩ 𝐵))) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))))
3		un4 3323	. . . 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))))
4		simp3 1001	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵)))
5	4	uneq1d 3316	. . . . . 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))))
6	5	uneq2d 3317	. . . . 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)))))
7		resundi 4959	. . . . . . 7 ⊢ (𝐹 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵)))
8		inundifss 3528	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴
9		ssres2 4973	. . . . . . . 8 ⊢ (((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴 → (𝐹 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ⊆ (𝐹 ↾ 𝐴))
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ (𝐹 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ⊆ (𝐹 ↾ 𝐴)
11	7, 10	eqsstrri 3216	. . . . . 6 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵))) ⊆ (𝐹 ↾ 𝐴)
12		resundi 4959	. . . . . . 7 ⊢ (𝐺 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴))) = ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))
13		incom 3355	. . . . . . . . . 10 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
14	13	uneq1i 3313	. . . . . . . . 9 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴)) = ((𝐵 ∩ 𝐴) ∪ (𝐵 ∖ 𝐴))
15		inundifss 3528	. . . . . . . . 9 ⊢ ((𝐵 ∩ 𝐴) ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵
16	14, 15	eqsstri 3215	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵
17		ssres2 4973	. . . . . . . 8 ⊢ (((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵 → (𝐺 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴))) ⊆ (𝐺 ↾ 𝐵))
18	16, 17	ax-mp 5	. . . . . . 7 ⊢ (𝐺 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴))) ⊆ (𝐺 ↾ 𝐵)
19	12, 18	eqsstrri 3216	. . . . . 6 ⊢ ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))) ⊆ (𝐺 ↾ 𝐵)
20		unss12 3335	. . . . . 6 ⊢ ((((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵))) ⊆ (𝐹 ↾ 𝐴) ∧ ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))) ⊆ (𝐺 ↾ 𝐵)) → (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵))) ∪ ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) ⊆ ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)))
21	11, 19, 20	mp2an 426	. . . . 5 ⊢ (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵))) ∪ ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) ⊆ ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵))
22	6, 21	eqsstrdi 3235	. . . 4 ⊢ ((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) ∪ (𝐺 ↾ 𝐵)))
23	3, 22	eqsstrrid 3230	. . 3 ⊢ ((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) ∪ (𝐺 ↾ 𝐵)))
24	2, 23	eqsstrrid 3230	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) ⊆ ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)))
25		fnresdm 5367	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
26		fnresdm 5367	. . . 4 ⊢ (𝐺 Fn 𝐵 → (𝐺 ↾ 𝐵) = 𝐺)
27		uneq12 3312	. . . 4 ⊢ (((𝐹 ↾ 𝐴) = 𝐹 ∧ (𝐺 ↾ 𝐵) = 𝐺) → ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)) = (𝐹 ∪ 𝐺))
28	25, 26, 27	syl2an 289	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)) = (𝐹 ∪ 𝐺))
29	28	3adant3 1019	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)) = (𝐹 ∪ 𝐺))
30	24, 29	sseqtrd 3221	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) ⊆ (𝐹 ∪ 𝐺))

Syntax hints:   → wi 4   ∧ w3a 980   = wceq 1364   ∖ cdif 3154   ∪ cun 3155   ∩ cin 3156   ⊆ wss 3157   ↾ cres 4665   Fn wfn 5253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-dm 4673  df-res 4675  df-fun 5260  df-fn 5261

This theorem is referenced by: (None)