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Theorem fnfco 5964
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 5791 . 2 (𝐺:𝐵𝐴 ↔ (𝐺 Fn 𝐵 ∧ ran 𝐺𝐴))
2 fnco 5896 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → (𝐹𝐺) Fn 𝐵)
323expb 1257 . 2 ((𝐹 Fn 𝐴 ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺𝐴)) → (𝐹𝐺) Fn 𝐵)
41, 3sylan2b 490 1 ((𝐹 Fn 𝐴𝐺:𝐵𝐴) → (𝐹𝐺) Fn 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wss 3536  ran crn 5026  ccom 5029   Fn wfn 5782  wf 5783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pr 4825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-br 4575  df-opab 4635  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-fun 5789  df-fn 5790  df-f 5791
This theorem is referenced by:  cocan1  6421  cocan2  6422  ofco  6789  1stcof  7061  2ndcof  7062  axcc3  9117  dmaf  16465  cdaf  16466  gsumzaddlem  18087  prdstopn  21180  xpstopnlem2  21363  prdstgpd  21677  prdsxmslem2  22082  uniiccdif  23066  uniiccvol  23068  uniioombllem2  23071  resinf1o  24000  jensen  24429  occllem  27349  nlelchi  28107  hmopidmchi  28197  iprodefisumlem  30682  brcoffn  37148  brcofffn  37149  stoweidlem27  38721
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