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Theorem fcod 6743
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 6741 . 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 5676  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-11 2147  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-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  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-res 5684  df-ima 5685  df-fun 6544  df-fn 6545  df-f 6546
This theorem is referenced by:  suppcoss  8206  mapen  9159  mapfienlem3  9424  mapfien  9425  cofsmo  10286  canthp1lem2  10670  gsumval3lem2  19854  psrass1lem  21870  psdmplcl  22079  comet  24415  dvcobr  25870  gsumpart  32763  subfacp1lem5  34788  metakunt33  41683  mapcod  41727  mhmcompl  41775  selvvvval  41812  itcovalendof  47736
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