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Theorem fun2 6105
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 6104 . 2 (((𝐹:𝐴𝐶𝐺:𝐵𝐶) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐶))
2 unidm 3789 . . 3 (𝐶𝐶) = 𝐶
3 feq3 6066 . . 3 ((𝐶𝐶) = 𝐶 → ((𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐶) ↔ (𝐹𝐺):(𝐴𝐵)⟶𝐶))
42, 3ax-mp 5 . 2 ((𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐶) ↔ (𝐹𝐺):(𝐴𝐵)⟶𝐶)
51, 4sylib 208 1 (((𝐹:𝐴𝐶𝐺:𝐵𝐶) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  cun 3605  cin 3606  c0 3948  wf 5922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-fun 5928  df-fn 5929  df-f 5930
This theorem is referenced by:  fun2d  6106  fresaun  6113  mapunen  8170  ac6sfi  8245  axdc3lem4  9313  fseq1p1m1  12452  axlowdimlem5  25871  axlowdimlem7  25873  resf1o  29633  locfinref  30036  breprexplema  30836
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