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Theorem fun2 6786
Description: The union of two functions with disjoint domains. (Contributed by Mario Carneiro, 12-Mar-2015.)
Assertion
Ref Expression
fun2 (((𝐹:𝐴𝐶𝐺:𝐵𝐶) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)

Proof of Theorem fun2
StepHypRef Expression
1 fun 6785 . 2 (((𝐹:𝐴𝐶𝐺:𝐵𝐶) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐶))
2 unidm 4180 . . 3 (𝐶𝐶) = 𝐶
3 feq3 6732 . . 3 ((𝐶𝐶) = 𝐶 → ((𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐶) ↔ (𝐹𝐺):(𝐴𝐵)⟶𝐶))
42, 3ax-mp 5 . 2 ((𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐶) ↔ (𝐹𝐺):(𝐴𝐵)⟶𝐶)
51, 4sylib 218 1 (((𝐹:𝐴𝐶𝐺:𝐵𝐶) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  cun 3974  cin 3975  c0 4352  wf 6571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-fun 6577  df-fn 6578  df-f 6579
This theorem is referenced by:  fun2d  6787  axlowdimlem5  28981  axlowdimlem7  28983  resf1o  32746  locfinref  33789  breprexplema  34609
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