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Theorem fnfco 5008
Description: Composition of two functions. (Contributed by NM, 22-May-2006.)
Assertion
Ref Expression
fnfco ((𝐹 Fn A 𝐺:BA) → (𝐹𝐺) Fn B)

Proof of Theorem fnfco
StepHypRef Expression
1 df-f 4849 . 2 (𝐺:BA ↔ (𝐺 Fn B ran 𝐺A))
2 fnco 4950 . . 3 ((𝐹 Fn A 𝐺 Fn B ran 𝐺A) → (𝐹𝐺) Fn B)
323expb 1104 . 2 ((𝐹 Fn A (𝐺 Fn B ran 𝐺A)) → (𝐹𝐺) Fn B)
41, 3sylan2b 271 1 ((𝐹 Fn A 𝐺:BA) → (𝐹𝐺) Fn B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wss 2911  ran crn 4289  ccom 4292   Fn wfn 4840  wf 4841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849
This theorem is referenced by:  cocan1  5370  cocan2  5371  ofco  5671  1stcof  5732  2ndcof  5733
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