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Theorem funcofd 6702
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 (πœ‘ β†’ Fun 𝐹)
funcofd.2 (πœ‘ β†’ Fun 𝐺)
Assertion
Ref Expression
funcofd (πœ‘ β†’ (𝐹 ∘ 𝐺):(◑𝐺 β€œ dom 𝐹)⟢ran 𝐹)

Proof of Theorem funcofd
StepHypRef Expression
1 funcofd.1 . . 3 (πœ‘ β†’ Fun 𝐹)
2 fdmrn 6701 . . 3 (Fun 𝐹 ↔ 𝐹:dom 𝐹⟢ran 𝐹)
31, 2sylib 217 . 2 (πœ‘ β†’ 𝐹:dom 𝐹⟢ran 𝐹)
4 funcofd.2 . 2 (πœ‘ β†’ Fun 𝐺)
5 fcof 6692 . 2 ((𝐹:dom 𝐹⟢ran 𝐹 ∧ Fun 𝐺) β†’ (𝐹 ∘ 𝐺):(◑𝐺 β€œ dom 𝐹)⟢ran 𝐹)
63, 4, 5syl2anc 585 1 (πœ‘ β†’ (𝐹 ∘ 𝐺):(◑𝐺 β€œ dom 𝐹)⟢ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4  β—‘ccnv 5633  dom cdm 5634  ran crn 5635   β€œ cima 5637   ∘ ccom 5638  Fun wfun 6491  βŸΆwf 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-11 2155  ax-12 2172  ax-ext 2704  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-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  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-res 5646  df-ima 5647  df-fun 6499  df-fn 6500  df-f 6501
This theorem is referenced by:  smfco  45129
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