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Theorem fun 6771
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 6683 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺) Fn (𝐴𝐵))
21expcom 413 . . . 4 ((𝐴𝐵) = ∅ → ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹𝐺) Fn (𝐴𝐵)))
3 rnun 6168 . . . . 5 ran (𝐹𝐺) = (ran 𝐹 ∪ ran 𝐺)
4 unss12 4198 . . . . 5 ((ran 𝐹𝐶 ∧ ran 𝐺𝐷) → (ran 𝐹 ∪ ran 𝐺) ⊆ (𝐶𝐷))
53, 4eqsstrid 4044 . . . 4 ((ran 𝐹𝐶 ∧ ran 𝐺𝐷) → ran (𝐹𝐺) ⊆ (𝐶𝐷))
62, 5anim12d1 610 . . 3 ((𝐴𝐵) = ∅ → (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (ran 𝐹𝐶 ∧ ran 𝐺𝐷)) → ((𝐹𝐺) Fn (𝐴𝐵) ∧ ran (𝐹𝐺) ⊆ (𝐶𝐷))))
7 df-f 6567 . . . . 5 (𝐹:𝐴𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶))
8 df-f 6567 . . . . 5 (𝐺:𝐵𝐷 ↔ (𝐺 Fn 𝐵 ∧ ran 𝐺𝐷))
97, 8anbi12i 628 . . . 4 ((𝐹:𝐴𝐶𝐺:𝐵𝐷) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐶) ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺𝐷)))
10 an4 656 . . . 4 (((𝐹 Fn 𝐴 ∧ ran 𝐹𝐶) ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺𝐷)) ↔ ((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (ran 𝐹𝐶 ∧ ran 𝐺𝐷)))
119, 10bitri 275 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐷) ↔ ((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (ran 𝐹𝐶 ∧ ran 𝐺𝐷)))
12 df-f 6567 . . 3 ((𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐷) ↔ ((𝐹𝐺) Fn (𝐴𝐵) ∧ ran (𝐹𝐺) ⊆ (𝐶𝐷)))
136, 11, 123imtr4g 296 . 2 ((𝐴𝐵) = ∅ → ((𝐹:𝐴𝐶𝐺:𝐵𝐷) → (𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐷)))
1413impcom 407 1 (((𝐹:𝐴𝐶𝐺:𝐵𝐷) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  cun 3961  cin 3962  wss 3963  c0 4339  ran crn 5690   Fn wfn 6558  wf 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567
This theorem is referenced by:  fun2  6772  ftpg  7176  fsnunf  7205  ralxpmap  8935  hashfxnn0  14373  cats1un  14756  pwssplit1  21076  axlowdimlem10  28981  wlkp1  29714  padct  32737  eulerpartlemt  34353  sseqf  34374  poimirlem3  37610  poimirlem16  37623  poimirlem19  37626  poimirlem22  37629  poimirlem23  37630  poimirlem24  37631  poimirlem25  37632  poimirlem28  37635  poimirlem29  37636  poimirlem31  37638  mapfzcons  42704  diophrw  42747  diophren  42801  pwssplit4  43078  aacllem  49032
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