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Theorem fnfco 5090
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 4930 . 2 (𝐺:𝐵𝐴 ↔ (𝐺 Fn 𝐵 ∧ ran 𝐺𝐴))
2 fnco 5032 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → (𝐹𝐺) Fn 𝐵)
323expb 1140 . 2 ((𝐹 Fn 𝐴 ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺𝐴)) → (𝐹𝐺) Fn 𝐵)
41, 3sylan2b 281 1 ((𝐹 Fn 𝐴𝐺:𝐵𝐴) → (𝐹𝐺) Fn 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wss 2974  ran crn 4366  ccom 4369   Fn wfn 4921  wf 4922
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-fun 4928  df-fn 4929  df-f 4930
This theorem is referenced by:  cocan1  5452  cocan2  5453  ofco  5754  1stcof  5815  2ndcof  5816
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