Theorem resasplitss 5089

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 3115	. . . 4 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∩ 𝐵))) = (𝐹 ↾ (𝐴 ∩ 𝐵))
2	1	uneq1i 3122	. . 3 ⊢ (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∩ 𝐵))) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))))
3		un4 3132	. . . 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 940	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵)))
5	4	uneq1d 3125	. . . . . 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 3126	. . . . 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 4643	. . . . . . 7 ⊢ (𝐹 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵)))
8		inundifss 3321	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴
9		ssres2 4656	. . . . . . . 8 ⊢ (((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴 → (𝐹 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ⊆ (𝐹 ↾ 𝐴))
10	8, 9	ax-mp 7	. . . . . . 7 ⊢ (𝐹 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ⊆ (𝐹 ↾ 𝐴)
11	7, 10	eqsstr3i 3030	. . . . . 6 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵))) ⊆ (𝐹 ↾ 𝐴)
12		resundi 4643	. . . . . . 7 ⊢ (𝐺 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴))) = ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))
13		incom 3158	. . . . . . . . . 10 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
14	13	uneq1i 3122	. . . . . . . . 9 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴)) = ((𝐵 ∩ 𝐴) ∪ (𝐵 ∖ 𝐴))
15		inundifss 3321	. . . . . . . . 9 ⊢ ((𝐵 ∩ 𝐴) ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵
16	14, 15	eqsstri 3029	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵
17		ssres2 4656	. . . . . . . 8 ⊢ (((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵 → (𝐺 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴))) ⊆ (𝐺 ↾ 𝐵))
18	16, 17	ax-mp 7	. . . . . . 7 ⊢ (𝐺 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴))) ⊆ (𝐺 ↾ 𝐵)
19	12, 18	eqsstr3i 3030	. . . . . 6 ⊢ ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))) ⊆ (𝐺 ↾ 𝐵)
20		unss12 3144	. . . . . 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 416	. . . . 5 ⊢ (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵))) ∪ ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) ⊆ ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵))
22	6, 21	syl6eqss 3049	. . . 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	syl5eqssr 3044	. . 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	syl5eqssr 3044	. 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 5028	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
26		fnresdm 5028	. . . 4 ⊢ (𝐺 Fn 𝐵 → (𝐺 ↾ 𝐵) = 𝐺)
27		uneq12 3121	. . . 4 ⊢ (((𝐹 ↾ 𝐴) = 𝐹 ∧ (𝐺 ↾ 𝐵) = 𝐺) → ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)) = (𝐹 ∪ 𝐺))
28	25, 26, 27	syl2an 283	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)) = (𝐹 ∪ 𝐺))
29	28	3adant3 958	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)) = (𝐹 ∪ 𝐺))
30	24, 29	sseqtrd 3035	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 919   = wceq 1284   ∖ cdif 2970   ∪ cun 2971   ∩ cin 2972   ⊆ wss 2973   ↾ cres 4365   Fn wfn 4917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-dm 4373  df-res 4375  df-fun 4924  df-fn 4925

This theorem is referenced by: (None)