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Theorem fnfco 6210
Description: Composition of two functions. (Contributed by NM, 22-May-2006.)
Assertion
Ref Expression
fnfco ((𝐹 Fn 𝐴𝐺:𝐵𝐴) → (𝐹𝐺) Fn 𝐵)

Proof of Theorem fnfco
StepHypRef Expression
1 df-f 6034 . 2 (𝐺:𝐵𝐴 ↔ (𝐺 Fn 𝐵 ∧ ran 𝐺𝐴))
2 fnco 6138 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → (𝐹𝐺) Fn 𝐵)
323expb 1113 . 2 ((𝐹 Fn 𝐴 ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺𝐴)) → (𝐹𝐺) Fn 𝐵)
41, 3sylan2b 581 1 ((𝐹 Fn 𝐴𝐺:𝐵𝐴) → (𝐹𝐺) Fn 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wss 3723  ran crn 5251  ccom 5254   Fn wfn 6025  wf 6026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-fun 6032  df-fn 6033  df-f 6034
This theorem is referenced by:  cocan1  6692  cocan2  6693  ofco  7068  1stcof  7349  2ndcof  7350  axcc3  9466  dmaf  16906  cdaf  16907  gsumzaddlem  18528  prdstopn  21652  xpstopnlem2  21835  prdstgpd  22148  prdsxmslem2  22554  uniiccdif  23566  uniiccvol  23568  uniioombllem2  23571  resinf1o  24503  jensen  24936  occllem  28502  nlelchi  29260  hmopidmchi  29350  iprodefisumlem  31964  brcoffn  38854  brcofffn  38855  stoweidlem27  40758
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