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Theorem fun2 6706
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 6705 . 2 (((𝐹:𝐴𝐶𝐺:𝐵𝐶) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐶))
2 unidm 4113 . . 3 (𝐶𝐶) = 𝐶
3 feq3 6652 . . 3 ((𝐶𝐶) = 𝐶 → ((𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐶) ↔ (𝐹𝐺):(𝐴𝐵)⟶𝐶))
42, 3ax-mp 5 . 2 ((𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐶) ↔ (𝐹𝐺):(𝐴𝐵)⟶𝐶)
51, 4sylib 217 1 (((𝐹:𝐴𝐶𝐺:𝐵𝐶) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  cun 3909  cin 3910  c0 4283  wf 6493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-fun 6499  df-fn 6500  df-f 6501
This theorem is referenced by:  fun2d  6707  axlowdimlem5  27898  axlowdimlem7  27900  resf1o  31650  locfinref  32425  breprexplema  33246
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