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Theorem fnun 5341
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 5238 . . 3 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
2 df-fn 5238 . . 3 (𝐺 Fn 𝐵 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐵))
3 ineq12 3346 . . . . . . . . . . 11 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
43eqeq1d 2198 . . . . . . . . . 10 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → ((dom 𝐹 ∩ dom 𝐺) = ∅ ↔ (𝐴𝐵) = ∅))
54anbi2d 464 . . . . . . . . 9 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) ↔ ((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴𝐵) = ∅)))
6 funun 5279 . . . . . . . . 9 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹𝐺))
75, 6biimtrrdi 164 . . . . . . . 8 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴𝐵) = ∅) → Fun (𝐹𝐺)))
8 dmun 4852 . . . . . . . . 9 dom (𝐹𝐺) = (dom 𝐹 ∪ dom 𝐺)
9 uneq12 3299 . . . . . . . . 9 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (dom 𝐹 ∪ dom 𝐺) = (𝐴𝐵))
108, 9eqtrid 2234 . . . . . . . 8 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → dom (𝐹𝐺) = (𝐴𝐵))
117, 10jctird 317 . . . . . . 7 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴𝐵) = ∅) → (Fun (𝐹𝐺) ∧ dom (𝐹𝐺) = (𝐴𝐵))))
12 df-fn 5238 . . . . . . 7 ((𝐹𝐺) Fn (𝐴𝐵) ↔ (Fun (𝐹𝐺) ∧ dom (𝐹𝐺) = (𝐴𝐵)))
1311, 12imbitrrdi 162 . . . . . 6 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺) Fn (𝐴𝐵)))
1413expd 258 . . . . 5 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐴𝐵) = ∅ → (𝐹𝐺) Fn (𝐴𝐵))))
1514impcom 125 . . . 4 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵)) → ((𝐴𝐵) = ∅ → (𝐹𝐺) Fn (𝐴𝐵)))
1615an4s 588 . . 3 (((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ∧ (Fun 𝐺 ∧ dom 𝐺 = 𝐵)) → ((𝐴𝐵) = ∅ → (𝐹𝐺) Fn (𝐴𝐵)))
171, 2, 16syl2anb 291 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((𝐴𝐵) = ∅ → (𝐹𝐺) Fn (𝐴𝐵)))
1817imp 124 1 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺) Fn (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  cun 3142  cin 3143  c0 3437  dom cdm 4644  Fun wfun 5229   Fn wfn 5230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-id 4311  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-fun 5237  df-fn 5238
This theorem is referenced by:  fnunsn  5342  fun  5407  foun  5499  f1oun  5500
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