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Theorem fnun 5106
Description: The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
Assertion
Ref Expression
fnun (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺) Fn (𝐴𝐵))

Proof of Theorem fnun
StepHypRef Expression
1 df-fn 5005 . . 3 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
2 df-fn 5005 . . 3 (𝐺 Fn 𝐵 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐵))
3 ineq12 3194 . . . . . . . . . . 11 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
43eqeq1d 2096 . . . . . . . . . 10 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → ((dom 𝐹 ∩ dom 𝐺) = ∅ ↔ (𝐴𝐵) = ∅))
54anbi2d 452 . . . . . . . . 9 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) ↔ ((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴𝐵) = ∅)))
6 funun 5044 . . . . . . . . 9 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹𝐺))
75, 6syl6bir 162 . . . . . . . 8 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴𝐵) = ∅) → Fun (𝐹𝐺)))
8 dmun 4631 . . . . . . . . 9 dom (𝐹𝐺) = (dom 𝐹 ∪ dom 𝐺)
9 uneq12 3147 . . . . . . . . 9 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (dom 𝐹 ∪ dom 𝐺) = (𝐴𝐵))
108, 9syl5eq 2132 . . . . . . . 8 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → dom (𝐹𝐺) = (𝐴𝐵))
117, 10jctird 310 . . . . . . 7 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴𝐵) = ∅) → (Fun (𝐹𝐺) ∧ dom (𝐹𝐺) = (𝐴𝐵))))
12 df-fn 5005 . . . . . . 7 ((𝐹𝐺) Fn (𝐴𝐵) ↔ (Fun (𝐹𝐺) ∧ dom (𝐹𝐺) = (𝐴𝐵)))
1311, 12syl6ibr 160 . . . . . 6 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺) Fn (𝐴𝐵)))
1413expd 254 . . . . 5 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐴𝐵) = ∅ → (𝐹𝐺) Fn (𝐴𝐵))))
1514impcom 123 . . . 4 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵)) → ((𝐴𝐵) = ∅ → (𝐹𝐺) Fn (𝐴𝐵)))
1615an4s 555 . . 3 (((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ∧ (Fun 𝐺 ∧ dom 𝐺 = 𝐵)) → ((𝐴𝐵) = ∅ → (𝐹𝐺) Fn (𝐴𝐵)))
171, 2, 16syl2anb 285 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((𝐴𝐵) = ∅ → (𝐹𝐺) Fn (𝐴𝐵)))
1817imp 122 1 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺) Fn (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  cun 2995  cin 2996  c0 3284  dom cdm 4428  Fun wfun 4996   Fn wfn 4997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-fun 5004  df-fn 5005
This theorem is referenced by:  fnunsn  5107  fun  5168  foun  5256  f1oun  5257
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