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Theorem fun2 6754
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 6753 . 2 (((𝐹:𝐴𝐶𝐺:𝐵𝐶) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐶))
2 unidm 4148 . . 3 (𝐶𝐶) = 𝐶
3 feq3 6699 . . 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 395   = wceq 1534  cun 3943  cin 3944  c0 4318  wf 6538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-fun 6544  df-fn 6545  df-f 6546
This theorem is referenced by:  fun2d  6755  axlowdimlem5  28750  axlowdimlem7  28752  resf1o  32506  locfinref  33436  breprexplema  34256
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