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Theorem fresaunres2 6114
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 6083 . . . 4 (𝐹:𝐴𝐶𝐹 Fn 𝐴)
2 ffn 6083 . . . 4 (𝐺:𝐵𝐶𝐺 Fn 𝐵)
3 id 22 . . . 4 ((𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)) → (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)))
4 resasplit 6112 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺) = ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))))
51, 2, 3, 4syl3an 1408 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺) = ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))))
65reseq1d 5427 . 2 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ↾ 𝐵) = (((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ↾ 𝐵))
7 resundir 5446 . . 3 (((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ↾ 𝐵) = (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) ↾ 𝐵))
8 inss2 3867 . . . . . 6 (𝐴𝐵) ⊆ 𝐵
9 resabs2 5464 . . . . . 6 ((𝐴𝐵) ⊆ 𝐵 → ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = (𝐹 ↾ (𝐴𝐵)))
108, 9ax-mp 5 . . . . 5 ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = (𝐹 ↾ (𝐴𝐵))
11 resundir 5446 . . . . 5 (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) ↾ 𝐵) = (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵))
1210, 11uneq12i 3798 . . . 4 (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) ↾ 𝐵)) = ((𝐹 ↾ (𝐴𝐵)) ∪ (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵)))
13 dmres 5454 . . . . . . . . 9 dom ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = (𝐵 ∩ dom (𝐹 ↾ (𝐴𝐵)))
14 dmres 5454 . . . . . . . . . . 11 dom (𝐹 ↾ (𝐴𝐵)) = ((𝐴𝐵) ∩ dom 𝐹)
1514ineq2i 3844 . . . . . . . . . 10 (𝐵 ∩ dom (𝐹 ↾ (𝐴𝐵))) = (𝐵 ∩ ((𝐴𝐵) ∩ dom 𝐹))
16 disjdif 4073 . . . . . . . . . . . 12 (𝐵 ∩ (𝐴𝐵)) = ∅
1716ineq1i 3843 . . . . . . . . . . 11 ((𝐵 ∩ (𝐴𝐵)) ∩ dom 𝐹) = (∅ ∩ dom 𝐹)
18 inass 3856 . . . . . . . . . . 11 ((𝐵 ∩ (𝐴𝐵)) ∩ dom 𝐹) = (𝐵 ∩ ((𝐴𝐵) ∩ dom 𝐹))
19 inss1 3866 . . . . . . . . . . . 12 (∅ ∩ dom 𝐹) ⊆ ∅
20 0ss 4005 . . . . . . . . . . . 12 ∅ ⊆ (∅ ∩ dom 𝐹)
2119, 20eqssi 3652 . . . . . . . . . . 11 (∅ ∩ dom 𝐹) = ∅
2217, 18, 213eqtr3i 2681 . . . . . . . . . 10 (𝐵 ∩ ((𝐴𝐵) ∩ dom 𝐹)) = ∅
2315, 22eqtri 2673 . . . . . . . . 9 (𝐵 ∩ dom (𝐹 ↾ (𝐴𝐵))) = ∅
2413, 23eqtri 2673 . . . . . . . 8 dom ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = ∅
25 relres 5461 . . . . . . . . 9 Rel ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵)
26 reldm0 5375 . . . . . . . . 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 221 . . . . . . 7 ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = ∅
29 difss 3770 . . . . . . . 8 (𝐵𝐴) ⊆ 𝐵
30 resabs2 5464 . . . . . . . 8 ((𝐵𝐴) ⊆ 𝐵 → ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵) = (𝐺 ↾ (𝐵𝐴)))
3129, 30ax-mp 5 . . . . . . 7 ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵) = (𝐺 ↾ (𝐵𝐴))
3228, 31uneq12i 3798 . . . . . 6 (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵)) = (∅ ∪ (𝐺 ↾ (𝐵𝐴)))
3332uneq2i 3797 . . . . 5 ((𝐹 ↾ (𝐴𝐵)) ∪ (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵))) = ((𝐹 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴))))
34 simp3 1083 . . . . . . 7 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)))
3534uneq1d 3799 . . . . . 6 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴)))) = ((𝐺 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴)))))
36 uncom 3790 . . . . . . . . . 10 (∅ ∪ (𝐺 ↾ (𝐵𝐴))) = ((𝐺 ↾ (𝐵𝐴)) ∪ ∅)
37 un0 4000 . . . . . . . . . 10 ((𝐺 ↾ (𝐵𝐴)) ∪ ∅) = (𝐺 ↾ (𝐵𝐴))
3836, 37eqtri 2673 . . . . . . . . 9 (∅ ∪ (𝐺 ↾ (𝐵𝐴))) = (𝐺 ↾ (𝐵𝐴))
3938uneq2i 3797 . . . . . . . 8 ((𝐺 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴)))) = ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))
40 resundi 5445 . . . . . . . . 9 (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) = ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))
41 incom 3838 . . . . . . . . . . . . 13 (𝐴𝐵) = (𝐵𝐴)
4241uneq1i 3796 . . . . . . . . . . . 12 ((𝐴𝐵) ∪ (𝐵𝐴)) = ((𝐵𝐴) ∪ (𝐵𝐴))
43 inundif 4079 . . . . . . . . . . . 12 ((𝐵𝐴) ∪ (𝐵𝐴)) = 𝐵
4442, 43eqtri 2673 . . . . . . . . . . 11 ((𝐴𝐵) ∪ (𝐵𝐴)) = 𝐵
4544reseq2i 5425 . . . . . . . . . 10 (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) = (𝐺𝐵)
46 fnresdm 6038 . . . . . . . . . . . 12 (𝐺 Fn 𝐵 → (𝐺𝐵) = 𝐺)
472, 46syl 17 . . . . . . . . . . 11 (𝐺:𝐵𝐶 → (𝐺𝐵) = 𝐺)
4847adantl 481 . . . . . . . . . 10 ((𝐹:𝐴𝐶𝐺:𝐵𝐶) → (𝐺𝐵) = 𝐺)
4945, 48syl5eq 2697 . . . . . . . . 9 ((𝐹:𝐴𝐶𝐺:𝐵𝐶) → (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) = 𝐺)
5040, 49syl5eqr 2699 . . . . . . . 8 ((𝐹:𝐴𝐶𝐺:𝐵𝐶) → ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) = 𝐺)
5139, 50syl5eq 2697 . . . . . . 7 ((𝐹:𝐴𝐶𝐺:𝐵𝐶) → ((𝐺 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴)))) = 𝐺)
52513adant3 1101 . . . . . 6 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐺 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴)))) = 𝐺)
5335, 52eqtrd 2685 . . . . 5 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴)))) = 𝐺)
5433, 53syl5eq 2697 . . . 4 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵))) = 𝐺)
5512, 54syl5eq 2697 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) ↾ 𝐵)) = 𝐺)
567, 55syl5eq 2697 . 2 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ↾ 𝐵) = 𝐺)
576, 56eqtrd 2685 1 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ↾ 𝐵) = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  cdif 3604  cun 3605  cin 3606  wss 3607  c0 3948  dom cdm 5143  cres 5145  Rel wrel 5148   Fn wfn 5921  wf 5922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-dm 5153  df-res 5155  df-fun 5928  df-fn 5929  df-f 5930
This theorem is referenced by:  fresaunres1  6115  mapunen  8170  ptuncnv  21658  cvmliftlem10  31402  elmapresaunres2  37652
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