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Theorem fun 5114
Description: The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
Assertion
Ref Expression
fun (((𝐹:𝐴𝐶𝐺:𝐵𝐷) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐷))

Proof of Theorem fun
StepHypRef Expression
1 fnun 5056 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺) Fn (𝐴𝐵))
21expcom 114 . . . 4 ((𝐴𝐵) = ∅ → ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹𝐺) Fn (𝐴𝐵)))
3 rnun 4782 . . . . . 6 ran (𝐹𝐺) = (ran 𝐹 ∪ ran 𝐺)
4 unss12 3154 . . . . . 6 ((ran 𝐹𝐶 ∧ ran 𝐺𝐷) → (ran 𝐹 ∪ ran 𝐺) ⊆ (𝐶𝐷))
53, 4syl5eqss 3052 . . . . 5 ((ran 𝐹𝐶 ∧ ran 𝐺𝐷) → ran (𝐹𝐺) ⊆ (𝐶𝐷))
65a1i 9 . . . 4 ((𝐴𝐵) = ∅ → ((ran 𝐹𝐶 ∧ ran 𝐺𝐷) → ran (𝐹𝐺) ⊆ (𝐶𝐷)))
72, 6anim12d 328 . . 3 ((𝐴𝐵) = ∅ → (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (ran 𝐹𝐶 ∧ ran 𝐺𝐷)) → ((𝐹𝐺) Fn (𝐴𝐵) ∧ ran (𝐹𝐺) ⊆ (𝐶𝐷))))
8 df-f 4956 . . . . 5 (𝐹:𝐴𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶))
9 df-f 4956 . . . . 5 (𝐺:𝐵𝐷 ↔ (𝐺 Fn 𝐵 ∧ ran 𝐺𝐷))
108, 9anbi12i 448 . . . 4 ((𝐹:𝐴𝐶𝐺:𝐵𝐷) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐶) ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺𝐷)))
11 an4 551 . . . 4 (((𝐹 Fn 𝐴 ∧ ran 𝐹𝐶) ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺𝐷)) ↔ ((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (ran 𝐹𝐶 ∧ ran 𝐺𝐷)))
1210, 11bitri 182 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐷) ↔ ((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (ran 𝐹𝐶 ∧ ran 𝐺𝐷)))
13 df-f 4956 . . 3 ((𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐷) ↔ ((𝐹𝐺) Fn (𝐴𝐵) ∧ ran (𝐹𝐺) ⊆ (𝐶𝐷)))
147, 12, 133imtr4g 203 . 2 ((𝐴𝐵) = ∅ → ((𝐹:𝐴𝐶𝐺:𝐵𝐷) → (𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐷)))
1514impcom 123 1 (((𝐹:𝐴𝐶𝐺:𝐵𝐷) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  cun 2980  cin 2981  wss 2982  c0 3267  ran crn 4392   Fn wfn 4947  wf 4948
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-id 4076  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-fun 4954  df-fn 4955  df-f 4956
This theorem is referenced by:  fun2  5115  ftpg  5400  fsnunf  5415
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