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Theorem resasplit 6779
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 6688 . . . 4 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
2 fnresdm 6688 . . . 4 (𝐺 Fn 𝐵 → (𝐺𝐵) = 𝐺)
3 uneq12 4173 . . . 4 (((𝐹𝐴) = 𝐹 ∧ (𝐺𝐵) = 𝐺) → ((𝐹𝐴) ∪ (𝐺𝐵)) = (𝐹𝐺))
41, 2, 3syl2an 596 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((𝐹𝐴) ∪ (𝐺𝐵)) = (𝐹𝐺))
543adant3 1131 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐴) ∪ (𝐺𝐵)) = (𝐹𝐺))
6 inundif 4485 . . . . . . . 8 ((𝐴𝐵) ∪ (𝐴𝐵)) = 𝐴
76reseq2i 5997 . . . . . . 7 (𝐹 ↾ ((𝐴𝐵) ∪ (𝐴𝐵))) = (𝐹𝐴)
8 resundi 6014 . . . . . . 7 (𝐹 ↾ ((𝐴𝐵) ∪ (𝐴𝐵))) = ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵)))
97, 8eqtr3i 2765 . . . . . 6 (𝐹𝐴) = ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵)))
10 incom 4217 . . . . . . . . . 10 (𝐴𝐵) = (𝐵𝐴)
1110uneq1i 4174 . . . . . . . . 9 ((𝐴𝐵) ∪ (𝐵𝐴)) = ((𝐵𝐴) ∪ (𝐵𝐴))
12 inundif 4485 . . . . . . . . 9 ((𝐵𝐴) ∪ (𝐵𝐴)) = 𝐵
1311, 12eqtri 2763 . . . . . . . 8 ((𝐴𝐵) ∪ (𝐵𝐴)) = 𝐵
1413reseq2i 5997 . . . . . . 7 (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) = (𝐺𝐵)
15 resundi 6014 . . . . . . 7 (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) = ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))
1614, 15eqtr3i 2765 . . . . . 6 (𝐺𝐵) = ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))
179, 16uneq12i 4176 . . . . 5 ((𝐹𝐴) ∪ (𝐺𝐵)) = (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))))
18 simp3 1137 . . . . . . 7 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)))
1918uneq1d 4177 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) = ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))))
2019uneq2d 4178 . . . . 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)))))
2117, 20eqtr4id 2794 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐴) ∪ (𝐺𝐵)) = (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))))
22 un4 4185 . . . 4 (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) = (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))))
2321, 22eqtrdi 2791 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐴) ∪ (𝐺𝐵)) = (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))))
24 unidm 4167 . . . 4 ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) = (𝐹 ↾ (𝐴𝐵))
2524uneq1i 4174 . . 3 (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) = ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))))
2623, 25eqtrdi 2791 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐴) ∪ (𝐺𝐵)) = ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))))
275, 26eqtr3d 2777 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 1086   = wceq 1537  cdif 3960  cun 3961  cin 3962  cres 5691   Fn wfn 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-dm 5699  df-res 5701  df-fun 6565  df-fn 6566
This theorem is referenced by:  fresaun  6780  fresaunres2  6781
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