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Theorem fcod 6749
Description: Composition of two mappings. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
fcod.1 (𝜑𝐹:𝐵𝐶)
fcod.2 (𝜑𝐺:𝐴𝐵)
Assertion
Ref Expression
fcod (𝜑 → (𝐹𝐺):𝐴𝐶)

Proof of Theorem fcod
StepHypRef Expression
1 fcod.1 . 2 (𝜑𝐹:𝐵𝐶)
2 fcod.2 . 2 (𝜑𝐺:𝐴𝐵)
3 fco 6747 . 2 ((𝐹:𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)
41, 2, 3syl2anc 583 1 (𝜑 → (𝐹𝐺):𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  ccom 5682  wf 6544
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-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
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-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-fun 6550  df-fn 6551  df-f 6552
This theorem is referenced by:  suppcoss  8213  mapen  9166  mapfienlem3  9431  mapfien  9432  cofsmo  10293  canthp1lem2  10677  gsumval3lem2  19861  psrass1lem  21877  psdmplcl  22086  comet  24435  dvcobr  25890  gsumpart  32782  subfacp1lem5  34794  metakunt33  41689  mapcod  41733  mhmcompl  41781  selvvvval  41818  itcovalendof  47742
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