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Theorem resasplit 6542
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 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺) = ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))))

Proof of Theorem resasplit
StepHypRef Expression
1 fnresdm 6460 . . . 4 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
2 fnresdm 6460 . . . 4 (𝐺 Fn 𝐵 → (𝐺𝐵) = 𝐺)
3 uneq12 4133 . . . 4 (((𝐹𝐴) = 𝐹 ∧ (𝐺𝐵) = 𝐺) → ((𝐹𝐴) ∪ (𝐺𝐵)) = (𝐹𝐺))
41, 2, 3syl2an 595 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((𝐹𝐴) ∪ (𝐺𝐵)) = (𝐹𝐺))
543adant3 1124 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐴) ∪ (𝐺𝐵)) = (𝐹𝐺))
6 simp3 1130 . . . . . . 7 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)))
76uneq1d 4137 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) = ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))))
87uneq2d 4138 . . . . 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)))))
9 inundif 4425 . . . . . . . 8 ((𝐴𝐵) ∪ (𝐴𝐵)) = 𝐴
109reseq2i 5844 . . . . . . 7 (𝐹 ↾ ((𝐴𝐵) ∪ (𝐴𝐵))) = (𝐹𝐴)
11 resundi 5861 . . . . . . 7 (𝐹 ↾ ((𝐴𝐵) ∪ (𝐴𝐵))) = ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵)))
1210, 11eqtr3i 2846 . . . . . 6 (𝐹𝐴) = ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵)))
13 incom 4177 . . . . . . . . . 10 (𝐴𝐵) = (𝐵𝐴)
1413uneq1i 4134 . . . . . . . . 9 ((𝐴𝐵) ∪ (𝐵𝐴)) = ((𝐵𝐴) ∪ (𝐵𝐴))
15 inundif 4425 . . . . . . . . 9 ((𝐵𝐴) ∪ (𝐵𝐴)) = 𝐵
1614, 15eqtri 2844 . . . . . . . 8 ((𝐴𝐵) ∪ (𝐵𝐴)) = 𝐵
1716reseq2i 5844 . . . . . . 7 (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) = (𝐺𝐵)
18 resundi 5861 . . . . . . 7 (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) = ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))
1917, 18eqtr3i 2846 . . . . . 6 (𝐺𝐵) = ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))
2012, 19uneq12i 4136 . . . . 5 ((𝐹𝐴) ∪ (𝐺𝐵)) = (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))))
218, 20syl6reqr 2875 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐴) ∪ (𝐺𝐵)) = (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))))
22 un4 4144 . . . 4 (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) = (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))))
2321, 22syl6eq 2872 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐴) ∪ (𝐺𝐵)) = (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))))
24 unidm 4127 . . . 4 ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) = (𝐹 ↾ (𝐴𝐵))
2524uneq1i 4134 . . 3 (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) = ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))))
2623, 25syl6eq 2872 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐴) ∪ (𝐺𝐵)) = ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))))
275, 26eqtr3d 2858 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺) = ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1079   = wceq 1528  cdif 3932  cun 3933  cin 3934  cres 5551   Fn wfn 6344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5059  df-opab 5121  df-xp 5555  df-rel 5556  df-dm 5559  df-res 5561  df-fun 6351  df-fn 6352
This theorem is referenced by:  fresaun  6543  fresaunres2  6544
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