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Theorem fresaun 6534
 Description: The union of two functions which agree on their common domain is a function. (Contributed by Stefan O'Rear, 9-Oct-2014.)
Assertion
Ref Expression
fresaun ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)

Proof of Theorem fresaun
StepHypRef Expression
1 simp1 1133 . . . 4 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → 𝐹:𝐴𝐶)
2 inss1 4133 . . . 4 (𝐴𝐵) ⊆ 𝐴
3 fssres 6529 . . . 4 ((𝐹:𝐴𝐶 ∧ (𝐴𝐵) ⊆ 𝐴) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)⟶𝐶)
41, 2, 3sylancl 589 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)⟶𝐶)
5 difss 4037 . . . . 5 (𝐴𝐵) ⊆ 𝐴
6 fssres 6529 . . . . 5 ((𝐹:𝐴𝐶 ∧ (𝐴𝐵) ⊆ 𝐴) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)⟶𝐶)
71, 5, 6sylancl 589 . . . 4 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)⟶𝐶)
8 simp2 1134 . . . . 5 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → 𝐺:𝐵𝐶)
9 difss 4037 . . . . 5 (𝐵𝐴) ⊆ 𝐵
10 fssres 6529 . . . . 5 ((𝐺:𝐵𝐶 ∧ (𝐵𝐴) ⊆ 𝐵) → (𝐺 ↾ (𝐵𝐴)):(𝐵𝐴)⟶𝐶)
118, 9, 10sylancl 589 . . . 4 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐺 ↾ (𝐵𝐴)):(𝐵𝐴)⟶𝐶)
12 indifdir 4189 . . . . . 6 ((𝐴𝐵) ∩ (𝐵𝐴)) = ((𝐴 ∩ (𝐵𝐴)) ∖ (𝐵 ∩ (𝐵𝐴)))
13 disjdif 4368 . . . . . . 7 (𝐴 ∩ (𝐵𝐴)) = ∅
1413difeq1i 4024 . . . . . 6 ((𝐴 ∩ (𝐵𝐴)) ∖ (𝐵 ∩ (𝐵𝐴))) = (∅ ∖ (𝐵 ∩ (𝐵𝐴)))
15 0dif 4297 . . . . . 6 (∅ ∖ (𝐵 ∩ (𝐵𝐴))) = ∅
1612, 14, 153eqtri 2785 . . . . 5 ((𝐴𝐵) ∩ (𝐵𝐴)) = ∅
1716a1i 11 . . . 4 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐴𝐵) ∩ (𝐵𝐴)) = ∅)
187, 11, 17fun2d 6527 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))):((𝐴𝐵) ∪ (𝐵𝐴))⟶𝐶)
19 indi 4178 . . . . 5 ((𝐴𝐵) ∩ ((𝐴𝐵) ∪ (𝐵𝐴))) = (((𝐴𝐵) ∩ (𝐴𝐵)) ∪ ((𝐴𝐵) ∩ (𝐵𝐴)))
20 inass 4124 . . . . . . 7 ((𝐴𝐵) ∩ (𝐴𝐵)) = (𝐴 ∩ (𝐵 ∩ (𝐴𝐵)))
21 disjdif 4368 . . . . . . . 8 (𝐵 ∩ (𝐴𝐵)) = ∅
2221ineq2i 4114 . . . . . . 7 (𝐴 ∩ (𝐵 ∩ (𝐴𝐵))) = (𝐴 ∩ ∅)
23 in0 4287 . . . . . . 7 (𝐴 ∩ ∅) = ∅
2420, 22, 233eqtri 2785 . . . . . 6 ((𝐴𝐵) ∩ (𝐴𝐵)) = ∅
25 incom 4106 . . . . . . . 8 (𝐴𝐵) = (𝐵𝐴)
2625ineq1i 4113 . . . . . . 7 ((𝐴𝐵) ∩ (𝐵𝐴)) = ((𝐵𝐴) ∩ (𝐵𝐴))
27 inass 4124 . . . . . . . 8 ((𝐵𝐴) ∩ (𝐵𝐴)) = (𝐵 ∩ (𝐴 ∩ (𝐵𝐴)))
2813ineq2i 4114 . . . . . . . 8 (𝐵 ∩ (𝐴 ∩ (𝐵𝐴))) = (𝐵 ∩ ∅)
29 in0 4287 . . . . . . . 8 (𝐵 ∩ ∅) = ∅
3027, 28, 293eqtri 2785 . . . . . . 7 ((𝐵𝐴) ∩ (𝐵𝐴)) = ∅
3126, 30eqtri 2781 . . . . . 6 ((𝐴𝐵) ∩ (𝐵𝐴)) = ∅
3224, 31uneq12i 4066 . . . . 5 (((𝐴𝐵) ∩ (𝐴𝐵)) ∪ ((𝐴𝐵) ∩ (𝐵𝐴))) = (∅ ∪ ∅)
33 un0 4286 . . . . 5 (∅ ∪ ∅) = ∅
3419, 32, 333eqtri 2785 . . . 4 ((𝐴𝐵) ∩ ((𝐴𝐵) ∪ (𝐵𝐴))) = ∅
3534a1i 11 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐴𝐵) ∩ ((𝐴𝐵) ∪ (𝐵𝐴))) = ∅)
364, 18, 35fun2d 6527 . 2 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))):((𝐴𝐵) ∪ ((𝐴𝐵) ∪ (𝐵𝐴)))⟶𝐶)
37 un12 4072 . . . . 5 ((𝐴𝐵) ∪ ((𝐴𝐵) ∪ (𝐵𝐴))) = ((𝐴𝐵) ∪ ((𝐴𝐵) ∪ (𝐵𝐴)))
3825uneq1i 4064 . . . . . . 7 ((𝐴𝐵) ∪ (𝐵𝐴)) = ((𝐵𝐴) ∪ (𝐵𝐴))
39 inundif 4375 . . . . . . 7 ((𝐵𝐴) ∪ (𝐵𝐴)) = 𝐵
4038, 39eqtri 2781 . . . . . 6 ((𝐴𝐵) ∪ (𝐵𝐴)) = 𝐵
4140uneq2i 4065 . . . . 5 ((𝐴𝐵) ∪ ((𝐴𝐵) ∪ (𝐵𝐴))) = ((𝐴𝐵) ∪ 𝐵)
42 undif1 4372 . . . . 5 ((𝐴𝐵) ∪ 𝐵) = (𝐴𝐵)
4337, 41, 423eqtri 2785 . . . 4 ((𝐴𝐵) ∪ ((𝐴𝐵) ∪ (𝐵𝐴))) = (𝐴𝐵)
4443feq2i 6490 . . 3 (((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))):((𝐴𝐵) ∪ ((𝐴𝐵) ∪ (𝐵𝐴)))⟶𝐶 ↔ ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))):(𝐴𝐵)⟶𝐶)
45 ffn 6498 . . . . 5 (𝐹:𝐴𝐶𝐹 Fn 𝐴)
46 ffn 6498 . . . . 5 (𝐺:𝐵𝐶𝐺 Fn 𝐵)
47 id 22 . . . . 5 ((𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)) → (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)))
48 resasplit 6533 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺) = ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))))
4945, 46, 47, 48syl3an 1157 . . . 4 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺) = ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))))
5049feq1d 6483 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺):(𝐴𝐵)⟶𝐶 ↔ ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))):(𝐴𝐵)⟶𝐶))
5144, 50bitr4id 293 . 2 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))):((𝐴𝐵) ∪ ((𝐴𝐵) ∪ (𝐵𝐴)))⟶𝐶 ↔ (𝐹𝐺):(𝐴𝐵)⟶𝐶))
5236, 51mpbid 235 1 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∖ cdif 3855   ∪ cun 3856   ∩ cin 3857   ⊆ wss 3858  ∅c0 4225   ↾ cres 5526   Fn wfn 6330  ⟶wf 6331 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-fun 6337  df-fn 6338  df-f 6339 This theorem is referenced by:  elmapresaun  8462  cvmliftlem10  32772
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