MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fresaunres2 Structured version   Visualization version   GIF version

Theorem fresaunres2 6750
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 6704 . . . 4 (𝐹:𝐴𝐶𝐹 Fn 𝐴)
2 ffn 6704 . . . 4 (𝐺:𝐵𝐶𝐺 Fn 𝐵)
3 id 22 . . . 4 ((𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)) → (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)))
4 resasplit 6748 . . . 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 5972 . 2 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ↾ 𝐵) = (((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ↾ 𝐵))
7 resundir 5988 . . 3 (((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ↾ 𝐵) = (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) ↾ 𝐵))
8 inss2 4225 . . . . . 6 (𝐴𝐵) ⊆ 𝐵
9 resabs2 6005 . . . . . 6 ((𝐴𝐵) ⊆ 𝐵 → ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = (𝐹 ↾ (𝐴𝐵)))
108, 9ax-mp 5 . . . . 5 ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = (𝐹 ↾ (𝐴𝐵))
11 resundir 5988 . . . . 5 (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) ↾ 𝐵) = (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵))
1210, 11uneq12i 4157 . . . 4 (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) ↾ 𝐵)) = ((𝐹 ↾ (𝐴𝐵)) ∪ (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵)))
13 dmres 5995 . . . . . . . . 9 dom ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = (𝐵 ∩ dom (𝐹 ↾ (𝐴𝐵)))
14 dmres 5995 . . . . . . . . . . 11 dom (𝐹 ↾ (𝐴𝐵)) = ((𝐴𝐵) ∩ dom 𝐹)
1514ineq2i 4205 . . . . . . . . . 10 (𝐵 ∩ dom (𝐹 ↾ (𝐴𝐵))) = (𝐵 ∩ ((𝐴𝐵) ∩ dom 𝐹))
16 disjdif 4467 . . . . . . . . . . . 12 (𝐵 ∩ (𝐴𝐵)) = ∅
1716ineq1i 4204 . . . . . . . . . . 11 ((𝐵 ∩ (𝐴𝐵)) ∩ dom 𝐹) = (∅ ∩ dom 𝐹)
18 inass 4215 . . . . . . . . . . 11 ((𝐵 ∩ (𝐴𝐵)) ∩ dom 𝐹) = (𝐵 ∩ ((𝐴𝐵) ∩ dom 𝐹))
19 0in 4389 . . . . . . . . . . 11 (∅ ∩ dom 𝐹) = ∅
2017, 18, 193eqtr3i 2767 . . . . . . . . . 10 (𝐵 ∩ ((𝐴𝐵) ∩ dom 𝐹)) = ∅
2115, 20eqtri 2759 . . . . . . . . 9 (𝐵 ∩ dom (𝐹 ↾ (𝐴𝐵))) = ∅
2213, 21eqtri 2759 . . . . . . . 8 dom ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) = ∅
23 relres 6002 . . . . . . . . 9 Rel ((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵)
24 reldm0 5919 . . . . . . . . 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 4127 . . . . . . . 8 (𝐵𝐴) ⊆ 𝐵
28 resabs2 6005 . . . . . . . 8 ((𝐵𝐴) ⊆ 𝐵 → ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵) = (𝐺 ↾ (𝐵𝐴)))
2927, 28ax-mp 5 . . . . . . 7 ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵) = (𝐺 ↾ (𝐵𝐴))
3026, 29uneq12i 4157 . . . . . 6 (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵)) = (∅ ∪ (𝐺 ↾ (𝐵𝐴)))
3130uneq2i 4156 . . . . 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 4158 . . . . . 6 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴)))) = ((𝐺 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴)))))
34 uncom 4149 . . . . . . . . . 10 (∅ ∪ (𝐺 ↾ (𝐵𝐴))) = ((𝐺 ↾ (𝐵𝐴)) ∪ ∅)
35 un0 4386 . . . . . . . . . 10 ((𝐺 ↾ (𝐵𝐴)) ∪ ∅) = (𝐺 ↾ (𝐵𝐴))
3634, 35eqtri 2759 . . . . . . . . 9 (∅ ∪ (𝐺 ↾ (𝐵𝐴))) = (𝐺 ↾ (𝐵𝐴))
3736uneq2i 4156 . . . . . . . 8 ((𝐺 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴)))) = ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))
38 resundi 5987 . . . . . . . . 9 (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) = ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))
39 incom 4197 . . . . . . . . . . . . 13 (𝐴𝐵) = (𝐵𝐴)
4039uneq1i 4155 . . . . . . . . . . . 12 ((𝐴𝐵) ∪ (𝐵𝐴)) = ((𝐵𝐴) ∪ (𝐵𝐴))
41 inundif 4474 . . . . . . . . . . . 12 ((𝐵𝐴) ∪ (𝐵𝐴)) = 𝐵
4240, 41eqtri 2759 . . . . . . . . . . 11 ((𝐴𝐵) ∪ (𝐵𝐴)) = 𝐵
4342reseq2i 5970 . . . . . . . . . 10 (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) = (𝐺𝐵)
44 fnresdm 6656 . . . . . . . . . . . 12 (𝐺 Fn 𝐵 → (𝐺𝐵) = 𝐺)
452, 44syl 17 . . . . . . . . . . 11 (𝐺:𝐵𝐶 → (𝐺𝐵) = 𝐺)
4645adantl 482 . . . . . . . . . 10 ((𝐹:𝐴𝐶𝐺:𝐵𝐶) → (𝐺𝐵) = 𝐺)
4743, 46eqtrid 2783 . . . . . . . . 9 ((𝐹:𝐴𝐶𝐺:𝐵𝐶) → (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) = 𝐺)
4838, 47eqtr3id 2785 . . . . . . . 8 ((𝐹:𝐴𝐶𝐺:𝐵𝐶) → ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) = 𝐺)
4937, 48eqtrid 2783 . . . . . . 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 2771 . . . . 5 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ (∅ ∪ (𝐺 ↾ (𝐵𝐴)))) = 𝐺)
5231, 51eqtrid 2783 . . . 4 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ ((𝐺 ↾ (𝐵𝐴)) ↾ 𝐵))) = 𝐺)
5312, 52eqtrid 2783 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (((𝐹 ↾ (𝐴𝐵)) ↾ 𝐵) ∪ (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) ↾ 𝐵)) = 𝐺)
547, 53eqtrid 2783 . 2 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ↾ 𝐵) = 𝐺)
556, 54eqtrd 2771 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 3941  cun 3942  cin 3943  wss 3944  c0 4318  dom cdm 5669  cres 5671  Rel wrel 5674   Fn wfn 6527  wf 6528
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 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
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 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-dm 5679  df-res 5681  df-fun 6534  df-fn 6535  df-f 6536
This theorem is referenced by:  fresaunres1  6751  mapunen  9129  ptuncnv  23240  cvmliftlem10  34116  elmapresaunres2  41280
  Copyright terms: Public domain W3C validator