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Theorem fcod 6734
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 6732 . 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 5671  wf 6530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-fun 6536  df-fn 6537  df-f 6538
This theorem is referenced by:  suppcoss  8188  mapen  9138  mapfienlem3  9399  mapfien  9400  cofsmo  10261  canthp1lem2  10645  gsumval3lem2  19818  psrass1lem  21807  psdmplcl  22015  comet  24346  dvcobr  25801  gsumpart  32678  subfacp1lem5  34666  metakunt33  41514  mapcod  41564  mhmcompl  41613  selvvvval  41650  itcovalendof  47568
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