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Theorem funcofd 6633
Description: Composition of two functions as a function with domain and codomain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Proof shortened by AV, 20-Sep-2024.)
Hypotheses
Ref Expression
funcofd.1 (𝜑 → Fun 𝐹)
funcofd.2 (𝜑 → Fun 𝐺)
Assertion
Ref Expression
funcofd (𝜑 → (𝐹𝐺):(𝐺 “ dom 𝐹)⟶ran 𝐹)

Proof of Theorem funcofd
StepHypRef Expression
1 funcofd.1 . . 3 (𝜑 → Fun 𝐹)
2 fdmrn 6632 . . 3 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
31, 2sylib 217 . 2 (𝜑𝐹:dom 𝐹⟶ran 𝐹)
4 funcofd.2 . 2 (𝜑 → Fun 𝐺)
5 fcof 6623 . 2 ((𝐹:dom 𝐹⟶ran 𝐹 ∧ Fun 𝐺) → (𝐹𝐺):(𝐺 “ dom 𝐹)⟶ran 𝐹)
63, 4, 5syl2anc 584 1 (𝜑 → (𝐹𝐺):(𝐺 “ dom 𝐹)⟶ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  ccnv 5588  dom cdm 5589  ran crn 5590  cima 5592  ccom 5593  Fun wfun 6427  wf 6429
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-fun 6435  df-fn 6436  df-f 6437
This theorem is referenced by:  smfco  44336
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