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Theorem fun 5965
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 5897 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺) Fn (𝐴𝐵))
21expcom 449 . . . 4 ((𝐴𝐵) = ∅ → ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹𝐺) Fn (𝐴𝐵)))
3 rnun 5446 . . . . . 6 ran (𝐹𝐺) = (ran 𝐹 ∪ ran 𝐺)
4 unss12 3746 . . . . . 6 ((ran 𝐹𝐶 ∧ ran 𝐺𝐷) → (ran 𝐹 ∪ ran 𝐺) ⊆ (𝐶𝐷))
53, 4syl5eqss 3611 . . . . 5 ((ran 𝐹𝐶 ∧ ran 𝐺𝐷) → ran (𝐹𝐺) ⊆ (𝐶𝐷))
65a1i 11 . . . 4 ((𝐴𝐵) = ∅ → ((ran 𝐹𝐶 ∧ ran 𝐺𝐷) → ran (𝐹𝐺) ⊆ (𝐶𝐷)))
72, 6anim12d 583 . . 3 ((𝐴𝐵) = ∅ → (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (ran 𝐹𝐶 ∧ ran 𝐺𝐷)) → ((𝐹𝐺) Fn (𝐴𝐵) ∧ ran (𝐹𝐺) ⊆ (𝐶𝐷))))
8 df-f 5794 . . . . 5 (𝐹:𝐴𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶))
9 df-f 5794 . . . . 5 (𝐺:𝐵𝐷 ↔ (𝐺 Fn 𝐵 ∧ ran 𝐺𝐷))
108, 9anbi12i 728 . . . 4 ((𝐹:𝐴𝐶𝐺:𝐵𝐷) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐶) ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺𝐷)))
11 an4 860 . . . 4 (((𝐹 Fn 𝐴 ∧ ran 𝐹𝐶) ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺𝐷)) ↔ ((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (ran 𝐹𝐶 ∧ ran 𝐺𝐷)))
1210, 11bitri 262 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐷) ↔ ((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (ran 𝐹𝐶 ∧ ran 𝐺𝐷)))
13 df-f 5794 . . 3 ((𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐷) ↔ ((𝐹𝐺) Fn (𝐴𝐵) ∧ ran (𝐹𝐺) ⊆ (𝐶𝐷)))
147, 12, 133imtr4g 283 . 2 ((𝐴𝐵) = ∅ → ((𝐹:𝐴𝐶𝐺:𝐵𝐷) → (𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐷)))
1514impcom 444 1 (((𝐹:𝐴𝐶𝐺:𝐵𝐷) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  cun 3537  cin 3538  wss 3539  c0 3873  ran crn 5029   Fn wfn 5785  wf 5786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-id 4943  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-fun 5792  df-fn 5793  df-f 5794
This theorem is referenced by:  fun2  5966  ftpg  6306  fsnunf  6334  ralxpmap  7771  hashf  12944  cats1un  13276  pwssplit1  18829  axlowdimlem10  25577  constr3trllem3  25974  padct  28719  eulerpartlemt  29594  sseqf  29615  poimirlem3  32406  poimirlem16  32419  poimirlem19  32422  poimirlem22  32425  poimirlem23  32426  poimirlem24  32427  poimirlem25  32428  poimirlem28  32431  poimirlem29  32432  poimirlem31  32434  mapfzcons  36121  diophrw  36164  diophren  36219  pwssplit4  36501  1wlkp1  40912  aacllem  42339
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