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Theorem fun 5370
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 5304 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺) Fn (𝐴𝐵))
21expcom 115 . . . 4 ((𝐴𝐵) = ∅ → ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹𝐺) Fn (𝐴𝐵)))
3 rnun 5019 . . . . . 6 ran (𝐹𝐺) = (ran 𝐹 ∪ ran 𝐺)
4 unss12 3299 . . . . . 6 ((ran 𝐹𝐶 ∧ ran 𝐺𝐷) → (ran 𝐹 ∪ ran 𝐺) ⊆ (𝐶𝐷))
53, 4eqsstrid 3193 . . . . 5 ((ran 𝐹𝐶 ∧ ran 𝐺𝐷) → ran (𝐹𝐺) ⊆ (𝐶𝐷))
65a1i 9 . . . 4 ((𝐴𝐵) = ∅ → ((ran 𝐹𝐶 ∧ ran 𝐺𝐷) → ran (𝐹𝐺) ⊆ (𝐶𝐷)))
72, 6anim12d 333 . . 3 ((𝐴𝐵) = ∅ → (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (ran 𝐹𝐶 ∧ ran 𝐺𝐷)) → ((𝐹𝐺) Fn (𝐴𝐵) ∧ ran (𝐹𝐺) ⊆ (𝐶𝐷))))
8 df-f 5202 . . . . 5 (𝐹:𝐴𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶))
9 df-f 5202 . . . . 5 (𝐺:𝐵𝐷 ↔ (𝐺 Fn 𝐵 ∧ ran 𝐺𝐷))
108, 9anbi12i 457 . . . 4 ((𝐹:𝐴𝐶𝐺:𝐵𝐷) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐶) ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺𝐷)))
11 an4 581 . . . 4 (((𝐹 Fn 𝐴 ∧ ran 𝐹𝐶) ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺𝐷)) ↔ ((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (ran 𝐹𝐶 ∧ ran 𝐺𝐷)))
1210, 11bitri 183 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐷) ↔ ((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (ran 𝐹𝐶 ∧ ran 𝐺𝐷)))
13 df-f 5202 . . 3 ((𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐷) ↔ ((𝐹𝐺) Fn (𝐴𝐵) ∧ ran (𝐹𝐺) ⊆ (𝐶𝐷)))
147, 12, 133imtr4g 204 . 2 ((𝐴𝐵) = ∅ → ((𝐹:𝐴𝐶𝐺:𝐵𝐷) → (𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐷)))
1514impcom 124 1 (((𝐹:𝐴𝐶𝐺:𝐵𝐷) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  cun 3119  cin 3120  wss 3121  c0 3414  ran crn 4612   Fn wfn 5193  wf 5194
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-fun 5200  df-fn 5201  df-f 5202
This theorem is referenced by:  fun2  5371  ftpg  5680  fsnunf  5696
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