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Theorem funcofd 6547
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 6546 . . 3 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
31, 2sylib 221 . 2 (𝜑𝐹:dom 𝐹⟶ran 𝐹)
4 funcofd.2 . 2 (𝜑 → Fun 𝐺)
5 fcof 6537 . 2 ((𝐹:dom 𝐹⟶ran 𝐹 ∧ Fun 𝐺) → (𝐹𝐺):(𝐺 “ dom 𝐹)⟶ran 𝐹)
63, 4, 5syl2anc 587 1 (𝜑 → (𝐹𝐺):(𝐺 “ dom 𝐹)⟶ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  ccnv 5534  dom cdm 5535  ran crn 5536  cima 5538  ccom 5539  Fun wfun 6343  wf 6345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-br 5041  df-opab 5103  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-fun 6351  df-fn 6352  df-f 6353
This theorem is referenced by:  smfco  43915
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