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Theorem fresaunres2 6763
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 6717 . . . 4 (𝐹:𝐴𝐶𝐹 Fn 𝐴)
2 ffn 6717 . . . 4 (𝐺:𝐵𝐶𝐺 Fn 𝐵)
3 id 22 . . . 4 ((𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)) → (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)))
4 resasplit 6761 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺) = ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))))
51, 2, 3, 4syl3an 1160 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺) = ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))))
65reseq1d 5980 . 2 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ↾ 𝐵) = (((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ↾ 𝐵))
7 resundir 5996 . . 3 (((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ↾ 𝐵) = (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) ↾ 𝐵))
8 inss2 4229 . . . . . 6 (𝐴𝐵) ⊆ 𝐵
9 resabs2 6013 . . . . . 6 ((𝐴𝐵) ⊆ 𝐵 → ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = (𝐹 ↾ (𝐴𝐵)))
108, 9ax-mp 5 . . . . 5 ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = (𝐹 ↾ (𝐴𝐵))
11 resundir 5996 . . . . 5 (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) ↾ 𝐵) = (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵))
1210, 11uneq12i 4161 . . . 4 (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) ↾ 𝐵)) = ((𝐹 ↾ (𝐴𝐵)) ∪ (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵)))
13 dmres 6003 . . . . . . . . 9 dom ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = (𝐵 ∩ dom (𝐹 ↾ (𝐴𝐵)))
14 dmres 6003 . . . . . . . . . . 11 dom (𝐹 ↾ (𝐴𝐵)) = ((𝐴𝐵) ∩ dom 𝐹)
1514ineq2i 4209 . . . . . . . . . 10 (𝐵 ∩ dom (𝐹 ↾ (𝐴𝐵))) = (𝐵 ∩ ((𝐴𝐵) ∩ dom 𝐹))
16 disjdif 4471 . . . . . . . . . . . 12 (𝐵 ∩ (𝐴𝐵)) = ∅
1716ineq1i 4208 . . . . . . . . . . 11 ((𝐵 ∩ (𝐴𝐵)) ∩ dom 𝐹) = (∅ ∩ dom 𝐹)
18 inass 4219 . . . . . . . . . . 11 ((𝐵 ∩ (𝐴𝐵)) ∩ dom 𝐹) = (𝐵 ∩ ((𝐴𝐵) ∩ dom 𝐹))
19 0in 4393 . . . . . . . . . . 11 (∅ ∩ dom 𝐹) = ∅
2017, 18, 193eqtr3i 2768 . . . . . . . . . 10 (𝐵 ∩ ((𝐴𝐵) ∩ dom 𝐹)) = ∅
2115, 20eqtri 2760 . . . . . . . . 9 (𝐵 ∩ dom (𝐹 ↾ (𝐴𝐵))) = ∅
2213, 21eqtri 2760 . . . . . . . 8 dom ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = ∅
23 relres 6010 . . . . . . . . 9 Rel ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵)
24 reldm0 5927 . . . . . . . . 9 (Rel ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) → (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = ∅ ↔ dom ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = ∅))
2523, 24ax-mp 5 . . . . . . . 8 (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = ∅ ↔ dom ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = ∅)
2622, 25mpbir 230 . . . . . . 7 ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = ∅
27 difss 4131 . . . . . . . 8 (𝐵𝐴) ⊆ 𝐵
28 resabs2 6013 . . . . . . . 8 ((𝐵𝐴) ⊆ 𝐵 → ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵) = (𝐺 ↾ (𝐵𝐴)))
2927, 28ax-mp 5 . . . . . . 7 ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵) = (𝐺 ↾ (𝐵𝐴))
3026, 29uneq12i 4161 . . . . . 6 (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵)) = (∅ ∪ (𝐺 ↾ (𝐵𝐴)))
3130uneq2i 4160 . . . . 5 ((𝐹 ↾ (𝐴𝐵)) ∪ (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵))) = ((𝐹 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴))))
32 simp3 1138 . . . . . . 7 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)))
3332uneq1d 4162 . . . . . 6 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴)))) = ((𝐺 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴)))))
34 uncom 4153 . . . . . . . . . 10 (∅ ∪ (𝐺 ↾ (𝐵𝐴))) = ((𝐺 ↾ (𝐵𝐴)) ∪ ∅)
35 un0 4390 . . . . . . . . . 10 ((𝐺 ↾ (𝐵𝐴)) ∪ ∅) = (𝐺 ↾ (𝐵𝐴))
3634, 35eqtri 2760 . . . . . . . . 9 (∅ ∪ (𝐺 ↾ (𝐵𝐴))) = (𝐺 ↾ (𝐵𝐴))
3736uneq2i 4160 . . . . . . . 8 ((𝐺 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴)))) = ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))
38 resundi 5995 . . . . . . . . 9 (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) = ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))
39 incom 4201 . . . . . . . . . . . . 13 (𝐴𝐵) = (𝐵𝐴)
4039uneq1i 4159 . . . . . . . . . . . 12 ((𝐴𝐵) ∪ (𝐵𝐴)) = ((𝐵𝐴) ∪ (𝐵𝐴))
41 inundif 4478 . . . . . . . . . . . 12 ((𝐵𝐴) ∪ (𝐵𝐴)) = 𝐵
4240, 41eqtri 2760 . . . . . . . . . . 11 ((𝐴𝐵) ∪ (𝐵𝐴)) = 𝐵
4342reseq2i 5978 . . . . . . . . . 10 (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) = (𝐺𝐵)
44 fnresdm 6669 . . . . . . . . . . . 12 (𝐺 Fn 𝐵 → (𝐺𝐵) = 𝐺)
452, 44syl 17 . . . . . . . . . . 11 (𝐺:𝐵𝐶 → (𝐺𝐵) = 𝐺)
4645adantl 482 . . . . . . . . . 10 ((𝐹:𝐴𝐶𝐺:𝐵𝐶) → (𝐺𝐵) = 𝐺)
4743, 46eqtrid 2784 . . . . . . . . 9 ((𝐹:𝐴𝐶𝐺:𝐵𝐶) → (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) = 𝐺)
4838, 47eqtr3id 2786 . . . . . . . 8 ((𝐹:𝐴𝐶𝐺:𝐵𝐶) → ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) = 𝐺)
4937, 48eqtrid 2784 . . . . . . 7 ((𝐹:𝐴𝐶𝐺:𝐵𝐶) → ((𝐺 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴)))) = 𝐺)
50493adant3 1132 . . . . . 6 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐺 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴)))) = 𝐺)
5133, 50eqtrd 2772 . . . . 5 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴)))) = 𝐺)
5231, 51eqtrid 2784 . . . 4 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵))) = 𝐺)
5312, 52eqtrid 2784 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) ↾ 𝐵)) = 𝐺)
547, 53eqtrid 2784 . 2 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ↾ 𝐵) = 𝐺)
556, 54eqtrd 2772 1 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ↾ 𝐵) = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  cdif 3945  cun 3946  cin 3947  wss 3948  c0 4322  dom cdm 5676  cres 5678  Rel wrel 5681   Fn wfn 6538  wf 6539
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-dm 5686  df-res 5688  df-fun 6545  df-fn 6546  df-f 6547
This theorem is referenced by:  fresaunres1  6764  mapunen  9148  ptuncnv  23318  cvmliftlem10  34354  elmapresaunres2  41597
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