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Theorem fnun 4948
Description: The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
Assertion
Ref Expression
fnun (((𝐹 Fn A 𝐺 Fn B) (AB) = ∅) → (𝐹𝐺) Fn (AB))

Proof of Theorem fnun
StepHypRef Expression
1 df-fn 4848 . . 3 (𝐹 Fn A ↔ (Fun 𝐹 dom 𝐹 = A))
2 df-fn 4848 . . 3 (𝐺 Fn B ↔ (Fun 𝐺 dom 𝐺 = B))
3 ineq12 3127 . . . . . . . . . . 11 ((dom 𝐹 = A dom 𝐺 = B) → (dom 𝐹 ∩ dom 𝐺) = (AB))
43eqeq1d 2045 . . . . . . . . . 10 ((dom 𝐹 = A dom 𝐺 = B) → ((dom 𝐹 ∩ dom 𝐺) = ∅ ↔ (AB) = ∅))
54anbi2d 437 . . . . . . . . 9 ((dom 𝐹 = A dom 𝐺 = B) → (((Fun 𝐹 Fun 𝐺) (dom 𝐹 ∩ dom 𝐺) = ∅) ↔ ((Fun 𝐹 Fun 𝐺) (AB) = ∅)))
6 funun 4887 . . . . . . . . 9 (((Fun 𝐹 Fun 𝐺) (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹𝐺))
75, 6syl6bir 153 . . . . . . . 8 ((dom 𝐹 = A dom 𝐺 = B) → (((Fun 𝐹 Fun 𝐺) (AB) = ∅) → Fun (𝐹𝐺)))
8 dmun 4485 . . . . . . . . 9 dom (𝐹𝐺) = (dom 𝐹 ∪ dom 𝐺)
9 uneq12 3086 . . . . . . . . 9 ((dom 𝐹 = A dom 𝐺 = B) → (dom 𝐹 ∪ dom 𝐺) = (AB))
108, 9syl5eq 2081 . . . . . . . 8 ((dom 𝐹 = A dom 𝐺 = B) → dom (𝐹𝐺) = (AB))
117, 10jctird 300 . . . . . . 7 ((dom 𝐹 = A dom 𝐺 = B) → (((Fun 𝐹 Fun 𝐺) (AB) = ∅) → (Fun (𝐹𝐺) dom (𝐹𝐺) = (AB))))
12 df-fn 4848 . . . . . . 7 ((𝐹𝐺) Fn (AB) ↔ (Fun (𝐹𝐺) dom (𝐹𝐺) = (AB)))
1311, 12syl6ibr 151 . . . . . 6 ((dom 𝐹 = A dom 𝐺 = B) → (((Fun 𝐹 Fun 𝐺) (AB) = ∅) → (𝐹𝐺) Fn (AB)))
1413expd 245 . . . . 5 ((dom 𝐹 = A dom 𝐺 = B) → ((Fun 𝐹 Fun 𝐺) → ((AB) = ∅ → (𝐹𝐺) Fn (AB))))
1514impcom 116 . . . 4 (((Fun 𝐹 Fun 𝐺) (dom 𝐹 = A dom 𝐺 = B)) → ((AB) = ∅ → (𝐹𝐺) Fn (AB)))
1615an4s 522 . . 3 (((Fun 𝐹 dom 𝐹 = A) (Fun 𝐺 dom 𝐺 = B)) → ((AB) = ∅ → (𝐹𝐺) Fn (AB)))
171, 2, 16syl2anb 275 . 2 ((𝐹 Fn A 𝐺 Fn B) → ((AB) = ∅ → (𝐹𝐺) Fn (AB)))
1817imp 115 1 (((𝐹 Fn A 𝐺 Fn B) (AB) = ∅) → (𝐹𝐺) Fn (AB))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  cun 2909  cin 2910  c0 3218  dom cdm 4288  Fun wfun 4839   Fn wfn 4840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-fun 4847  df-fn 4848
This theorem is referenced by:  fnunsn  4949  fun  5006  foun  5088  f1oun  5089
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