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Theorem fnun 5199
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 5096 . . 3 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
2 df-fn 5096 . . 3 (𝐺 Fn 𝐵 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐵))
3 ineq12 3242 . . . . . . . . . . 11 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
43eqeq1d 2126 . . . . . . . . . 10 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → ((dom 𝐹 ∩ dom 𝐺) = ∅ ↔ (𝐴𝐵) = ∅))
54anbi2d 459 . . . . . . . . 9 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) ↔ ((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴𝐵) = ∅)))
6 funun 5137 . . . . . . . . 9 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹𝐺))
75, 6syl6bir 163 . . . . . . . 8 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴𝐵) = ∅) → Fun (𝐹𝐺)))
8 dmun 4716 . . . . . . . . 9 dom (𝐹𝐺) = (dom 𝐹 ∪ dom 𝐺)
9 uneq12 3195 . . . . . . . . 9 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (dom 𝐹 ∪ dom 𝐺) = (𝐴𝐵))
108, 9syl5eq 2162 . . . . . . . 8 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → dom (𝐹𝐺) = (𝐴𝐵))
117, 10jctird 315 . . . . . . 7 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴𝐵) = ∅) → (Fun (𝐹𝐺) ∧ dom (𝐹𝐺) = (𝐴𝐵))))
12 df-fn 5096 . . . . . . 7 ((𝐹𝐺) Fn (𝐴𝐵) ↔ (Fun (𝐹𝐺) ∧ dom (𝐹𝐺) = (𝐴𝐵)))
1311, 12syl6ibr 161 . . . . . 6 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺) Fn (𝐴𝐵)))
1413expd 256 . . . . 5 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐴𝐵) = ∅ → (𝐹𝐺) Fn (𝐴𝐵))))
1514impcom 124 . . . 4 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵)) → ((𝐴𝐵) = ∅ → (𝐹𝐺) Fn (𝐴𝐵)))
1615an4s 562 . . 3 (((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ∧ (Fun 𝐺 ∧ dom 𝐺 = 𝐵)) → ((𝐴𝐵) = ∅ → (𝐹𝐺) Fn (𝐴𝐵)))
171, 2, 16syl2anb 289 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((𝐴𝐵) = ∅ → (𝐹𝐺) Fn (𝐴𝐵)))
1817imp 123 1 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺) Fn (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  cun 3039  cin 3040  c0 3333  dom cdm 4509  Fun wfun 5087   Fn wfn 5088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-id 4185  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-fun 5095  df-fn 5096
This theorem is referenced by:  fnunsn  5200  fun  5265  foun  5354  f1oun  5355
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