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Theorem fresaunres2 5989
Description: From the union of two functions that agree on the domain overlap, either component can be recovered by restriction. (Contributed by Stefan O'Rear, 9-Oct-2014.)
Assertion
Ref Expression
fresaunres2 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ↾ 𝐵) = 𝐺)

Proof of Theorem fresaunres2
StepHypRef Expression
1 ffn 5958 . . . 4 (𝐹:𝐴𝐶𝐹 Fn 𝐴)
2 ffn 5958 . . . 4 (𝐺:𝐵𝐶𝐺 Fn 𝐵)
3 id 22 . . . 4 ((𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)) → (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)))
4 resasplit 5987 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺) = ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))))
51, 2, 3, 4syl3an 1360 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺) = ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))))
65reseq1d 5316 . 2 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ↾ 𝐵) = (((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ↾ 𝐵))
7 resundir 5331 . . 3 (((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ↾ 𝐵) = (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) ↾ 𝐵))
8 inss2 3796 . . . . . 6 (𝐴𝐵) ⊆ 𝐵
9 resabs2 5349 . . . . . 6 ((𝐴𝐵) ⊆ 𝐵 → ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = (𝐹 ↾ (𝐴𝐵)))
108, 9ax-mp 5 . . . . 5 ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = (𝐹 ↾ (𝐴𝐵))
11 resundir 5331 . . . . 5 (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) ↾ 𝐵) = (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵))
1210, 11uneq12i 3727 . . . 4 (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) ↾ 𝐵)) = ((𝐹 ↾ (𝐴𝐵)) ∪ (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵)))
13 dmres 5339 . . . . . . . . 9 dom ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = (𝐵 ∩ dom (𝐹 ↾ (𝐴𝐵)))
14 dmres 5339 . . . . . . . . . . 11 dom (𝐹 ↾ (𝐴𝐵)) = ((𝐴𝐵) ∩ dom 𝐹)
1514ineq2i 3773 . . . . . . . . . 10 (𝐵 ∩ dom (𝐹 ↾ (𝐴𝐵))) = (𝐵 ∩ ((𝐴𝐵) ∩ dom 𝐹))
16 disjdif 3992 . . . . . . . . . . . 12 (𝐵 ∩ (𝐴𝐵)) = ∅
1716ineq1i 3772 . . . . . . . . . . 11 ((𝐵 ∩ (𝐴𝐵)) ∩ dom 𝐹) = (∅ ∩ dom 𝐹)
18 inass 3785 . . . . . . . . . . 11 ((𝐵 ∩ (𝐴𝐵)) ∩ dom 𝐹) = (𝐵 ∩ ((𝐴𝐵) ∩ dom 𝐹))
19 inss1 3795 . . . . . . . . . . . 12 (∅ ∩ dom 𝐹) ⊆ ∅
20 0ss 3924 . . . . . . . . . . . 12 ∅ ⊆ (∅ ∩ dom 𝐹)
2119, 20eqssi 3584 . . . . . . . . . . 11 (∅ ∩ dom 𝐹) = ∅
2217, 18, 213eqtr3i 2640 . . . . . . . . . 10 (𝐵 ∩ ((𝐴𝐵) ∩ dom 𝐹)) = ∅
2315, 22eqtri 2632 . . . . . . . . 9 (𝐵 ∩ dom (𝐹 ↾ (𝐴𝐵))) = ∅
2413, 23eqtri 2632 . . . . . . . 8 dom ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = ∅
25 relres 5346 . . . . . . . . 9 Rel ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵)
26 reldm0 5264 . . . . . . . . 9 (Rel ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) → (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = ∅ ↔ dom ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = ∅))
2725, 26ax-mp 5 . . . . . . . 8 (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = ∅ ↔ dom ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = ∅)
2824, 27mpbir 220 . . . . . . 7 ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = ∅
29 difss 3699 . . . . . . . 8 (𝐵𝐴) ⊆ 𝐵
30 resabs2 5349 . . . . . . . 8 ((𝐵𝐴) ⊆ 𝐵 → ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵) = (𝐺 ↾ (𝐵𝐴)))
3129, 30ax-mp 5 . . . . . . 7 ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵) = (𝐺 ↾ (𝐵𝐴))
3228, 31uneq12i 3727 . . . . . 6 (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵)) = (∅ ∪ (𝐺 ↾ (𝐵𝐴)))
3332uneq2i 3726 . . . . 5 ((𝐹 ↾ (𝐴𝐵)) ∪ (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵))) = ((𝐹 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴))))
34 simp3 1056 . . . . . . 7 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)))
3534uneq1d 3728 . . . . . 6 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴)))) = ((𝐺 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴)))))
36 uncom 3719 . . . . . . . . . 10 (∅ ∪ (𝐺 ↾ (𝐵𝐴))) = ((𝐺 ↾ (𝐵𝐴)) ∪ ∅)
37 un0 3919 . . . . . . . . . 10 ((𝐺 ↾ (𝐵𝐴)) ∪ ∅) = (𝐺 ↾ (𝐵𝐴))
3836, 37eqtri 2632 . . . . . . . . 9 (∅ ∪ (𝐺 ↾ (𝐵𝐴))) = (𝐺 ↾ (𝐵𝐴))
3938uneq2i 3726 . . . . . . . 8 ((𝐺 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴)))) = ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))
40 resundi 5330 . . . . . . . . 9 (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) = ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))
41 incom 3767 . . . . . . . . . . . . 13 (𝐴𝐵) = (𝐵𝐴)
4241uneq1i 3725 . . . . . . . . . . . 12 ((𝐴𝐵) ∪ (𝐵𝐴)) = ((𝐵𝐴) ∪ (𝐵𝐴))
43 inundif 3998 . . . . . . . . . . . 12 ((𝐵𝐴) ∪ (𝐵𝐴)) = 𝐵
4442, 43eqtri 2632 . . . . . . . . . . 11 ((𝐴𝐵) ∪ (𝐵𝐴)) = 𝐵
4544reseq2i 5314 . . . . . . . . . 10 (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) = (𝐺𝐵)
46 fnresdm 5914 . . . . . . . . . . . 12 (𝐺 Fn 𝐵 → (𝐺𝐵) = 𝐺)
472, 46syl 17 . . . . . . . . . . 11 (𝐺:𝐵𝐶 → (𝐺𝐵) = 𝐺)
4847adantl 481 . . . . . . . . . 10 ((𝐹:𝐴𝐶𝐺:𝐵𝐶) → (𝐺𝐵) = 𝐺)
4945, 48syl5eq 2656 . . . . . . . . 9 ((𝐹:𝐴𝐶𝐺:𝐵𝐶) → (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) = 𝐺)
5040, 49syl5eqr 2658 . . . . . . . 8 ((𝐹:𝐴𝐶𝐺:𝐵𝐶) → ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) = 𝐺)
5139, 50syl5eq 2656 . . . . . . 7 ((𝐹:𝐴𝐶𝐺:𝐵𝐶) → ((𝐺 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴)))) = 𝐺)
52513adant3 1074 . . . . . 6 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐺 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴)))) = 𝐺)
5335, 52eqtrd 2644 . . . . 5 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴)))) = 𝐺)
5433, 53syl5eq 2656 . . . 4 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵))) = 𝐺)
5512, 54syl5eq 2656 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) ↾ 𝐵)) = 𝐺)
567, 55syl5eq 2656 . 2 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ↾ 𝐵) = 𝐺)
576, 56eqtrd 2644 1 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ↾ 𝐵) = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  cdif 3537  cun 3538  cin 3539  wss 3540  c0 3874  dom cdm 5038  cres 5040  Rel wrel 5043   Fn wfn 5799  wf 5800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-dm 5048  df-res 5050  df-fun 5806  df-fn 5807  df-f 5808
This theorem is referenced by:  fresaunres1  5990  mapunen  8014  ptuncnv  21420  cvmliftlem10  30530  elmapresaunres2  36353
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