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Theorem fun 5006
 Description: The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
Assertion
Ref Expression
fun (((𝐹:A𝐶 𝐺:B𝐷) (AB) = ∅) → (𝐹𝐺):(AB)⟶(𝐶𝐷))

Proof of Theorem fun
StepHypRef Expression
1 fnun 4948 . . . . 5 (((𝐹 Fn A 𝐺 Fn B) (AB) = ∅) → (𝐹𝐺) Fn (AB))
21expcom 109 . . . 4 ((AB) = ∅ → ((𝐹 Fn A 𝐺 Fn B) → (𝐹𝐺) Fn (AB)))
3 rnun 4675 . . . . . 6 ran (𝐹𝐺) = (ran 𝐹 ∪ ran 𝐺)
4 unss12 3109 . . . . . 6 ((ran 𝐹𝐶 ran 𝐺𝐷) → (ran 𝐹 ∪ ran 𝐺) ⊆ (𝐶𝐷))
53, 4syl5eqss 2983 . . . . 5 ((ran 𝐹𝐶 ran 𝐺𝐷) → ran (𝐹𝐺) ⊆ (𝐶𝐷))
65a1i 9 . . . 4 ((AB) = ∅ → ((ran 𝐹𝐶 ran 𝐺𝐷) → ran (𝐹𝐺) ⊆ (𝐶𝐷)))
72, 6anim12d 318 . . 3 ((AB) = ∅ → (((𝐹 Fn A 𝐺 Fn B) (ran 𝐹𝐶 ran 𝐺𝐷)) → ((𝐹𝐺) Fn (AB) ran (𝐹𝐺) ⊆ (𝐶𝐷))))
8 df-f 4849 . . . . 5 (𝐹:A𝐶 ↔ (𝐹 Fn A ran 𝐹𝐶))
9 df-f 4849 . . . . 5 (𝐺:B𝐷 ↔ (𝐺 Fn B ran 𝐺𝐷))
108, 9anbi12i 433 . . . 4 ((𝐹:A𝐶 𝐺:B𝐷) ↔ ((𝐹 Fn A ran 𝐹𝐶) (𝐺 Fn B ran 𝐺𝐷)))
11 an4 520 . . . 4 (((𝐹 Fn A ran 𝐹𝐶) (𝐺 Fn B ran 𝐺𝐷)) ↔ ((𝐹 Fn A 𝐺 Fn B) (ran 𝐹𝐶 ran 𝐺𝐷)))
1210, 11bitri 173 . . 3 ((𝐹:A𝐶 𝐺:B𝐷) ↔ ((𝐹 Fn A 𝐺 Fn B) (ran 𝐹𝐶 ran 𝐺𝐷)))
13 df-f 4849 . . 3 ((𝐹𝐺):(AB)⟶(𝐶𝐷) ↔ ((𝐹𝐺) Fn (AB) ran (𝐹𝐺) ⊆ (𝐶𝐷)))
147, 12, 133imtr4g 194 . 2 ((AB) = ∅ → ((𝐹:A𝐶 𝐺:B𝐷) → (𝐹𝐺):(AB)⟶(𝐶𝐷)))
1514impcom 116 1 (((𝐹:A𝐶 𝐺:B𝐷) (AB) = ∅) → (𝐹𝐺):(AB)⟶(𝐶𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∪ cun 2909   ∩ cin 2910   ⊆ wss 2911  ∅c0 3218  ran crn 4289   Fn wfn 4840  ⟶wf 4841 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-rn 4299  df-fun 4847  df-fn 4848  df-f 4849 This theorem is referenced by:  fun2  5007  ftpg  5290  fsnunf  5305
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