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Theorem resasplitss 5136
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 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ⊆ (𝐹𝐺))

Proof of Theorem resasplitss
StepHypRef Expression
1 unidm 3127 . . . 4 ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) = (𝐹 ↾ (𝐴𝐵))
21uneq1i 3134 . . 3 (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) = ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))))
3 un4 3144 . . . 4 (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) = (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))))
4 simp3 941 . . . . . . 7 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)))
54uneq1d 3137 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) = ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))))
65uneq2d 3138 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) = (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))))
7 resundi 4682 . . . . . . 7 (𝐹 ↾ ((𝐴𝐵) ∪ (𝐴𝐵))) = ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵)))
8 inundifss 3342 . . . . . . . 8 ((𝐴𝐵) ∪ (𝐴𝐵)) ⊆ 𝐴
9 ssres2 4695 . . . . . . . 8 (((𝐴𝐵) ∪ (𝐴𝐵)) ⊆ 𝐴 → (𝐹 ↾ ((𝐴𝐵) ∪ (𝐴𝐵))) ⊆ (𝐹𝐴))
108, 9ax-mp 7 . . . . . . 7 (𝐹 ↾ ((𝐴𝐵) ∪ (𝐴𝐵))) ⊆ (𝐹𝐴)
117, 10eqsstr3i 3041 . . . . . 6 ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ⊆ (𝐹𝐴)
12 resundi 4682 . . . . . . 7 (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) = ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))
13 incom 3176 . . . . . . . . . 10 (𝐴𝐵) = (𝐵𝐴)
1413uneq1i 3134 . . . . . . . . 9 ((𝐴𝐵) ∪ (𝐵𝐴)) = ((𝐵𝐴) ∪ (𝐵𝐴))
15 inundifss 3342 . . . . . . . . 9 ((𝐵𝐴) ∪ (𝐵𝐴)) ⊆ 𝐵
1614, 15eqsstri 3040 . . . . . . . 8 ((𝐴𝐵) ∪ (𝐵𝐴)) ⊆ 𝐵
17 ssres2 4695 . . . . . . . 8 (((𝐴𝐵) ∪ (𝐵𝐴)) ⊆ 𝐵 → (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) ⊆ (𝐺𝐵))
1816, 17ax-mp 7 . . . . . . 7 (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) ⊆ (𝐺𝐵)
1912, 18eqsstr3i 3041 . . . . . 6 ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) ⊆ (𝐺𝐵)
20 unss12 3156 . . . . . 6 ((((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ⊆ (𝐹𝐴) ∧ ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) ⊆ (𝐺𝐵)) → (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ⊆ ((𝐹𝐴) ∪ (𝐺𝐵)))
2111, 19, 20mp2an 417 . . . . 5 (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ⊆ ((𝐹𝐴) ∪ (𝐺𝐵))
226, 21syl6eqss 3060 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ⊆ ((𝐹𝐴) ∪ (𝐺𝐵)))
233, 22syl5eqssr 3055 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ⊆ ((𝐹𝐴) ∪ (𝐺𝐵)))
242, 23syl5eqssr 3055 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ⊆ ((𝐹𝐴) ∪ (𝐺𝐵)))
25 fnresdm 5074 . . . 4 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
26 fnresdm 5074 . . . 4 (𝐺 Fn 𝐵 → (𝐺𝐵) = 𝐺)
27 uneq12 3133 . . . 4 (((𝐹𝐴) = 𝐹 ∧ (𝐺𝐵) = 𝐺) → ((𝐹𝐴) ∪ (𝐺𝐵)) = (𝐹𝐺))
2825, 26, 27syl2an 283 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((𝐹𝐴) ∪ (𝐺𝐵)) = (𝐹𝐺))
29283adant3 959 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐴) ∪ (𝐺𝐵)) = (𝐹𝐺))
3024, 29sseqtrd 3046 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ⊆ (𝐹𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 920   = wceq 1285  cdif 2981  cun 2982  cin 2983  wss 2984  cres 4401   Fn wfn 4962
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-xp 4405  df-rel 4406  df-dm 4409  df-res 4411  df-fun 4969  df-fn 4970
This theorem is referenced by: (None)
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