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Theorem fnun 5197
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 5094 . . 3 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
2 df-fn 5094 . . 3 (𝐺 Fn 𝐵 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐵))
3 ineq12 3240 . . . . . . . . . . 11 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
43eqeq1d 2124 . . . . . . . . . 10 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → ((dom 𝐹 ∩ dom 𝐺) = ∅ ↔ (𝐴𝐵) = ∅))
54anbi2d 457 . . . . . . . . 9 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) ↔ ((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴𝐵) = ∅)))
6 funun 5135 . . . . . . . . 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 4714 . . . . . . . . 9 dom (𝐹𝐺) = (dom 𝐹 ∪ dom 𝐺)
9 uneq12 3193 . . . . . . . . 9 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (dom 𝐹 ∪ dom 𝐺) = (𝐴𝐵))
108, 9syl5eq 2160 . . . . . . . 8 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → dom (𝐹𝐺) = (𝐴𝐵))
117, 10jctird 313 . . . . . . 7 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴𝐵) = ∅) → (Fun (𝐹𝐺) ∧ dom (𝐹𝐺) = (𝐴𝐵))))
12 df-fn 5094 . . . . . . 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 560 . . 3 (((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ∧ (Fun 𝐺 ∧ dom 𝐺 = 𝐵)) → ((𝐴𝐵) = ∅ → (𝐹𝐺) Fn (𝐴𝐵)))
171, 2, 16syl2anb 287 . 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 1314  cun 3037  cin 3038  c0 3331  dom cdm 4507  Fun wfun 5085   Fn wfn 5086
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-id 4183  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-fun 5093  df-fn 5094
This theorem is referenced by:  fnunsn  5198  fun  5263  foun  5352  f1oun  5353
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