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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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