Theorem fnfco 3639

Description: Composition of two functions.

Assertion

Ref	Expression
fnfco	|- ((F Fn A /\ G:B-->A) -> (F o. G) Fn B)

Proof of Theorem fnfco

Step	Hyp	Ref	Expression
1		fnco 3592	. . 3 |- ((F Fn A /\ G Fn B /\ ran G (_ A) -> (F o. G) Fn B)
2	1	3expb 833	. 2 |- ((F Fn A /\ (G Fn B /\ ran G (_ A)) -> (F o. G) Fn B)
3		df-f 3191	. 2 |- (G:B-->A <-> (G Fn B /\ ran G (_ A))
4	2, 3	sylan2b 452	1 |- ((F Fn A /\ G:B-->A) -> (F o. G) Fn B)

Syntax hints:   -> wi 3   /\ wa 223   (_ wss 2045  ran crn 3168   o. ccom 3171   Fn wfn 3174  -->wf 3175

This theorem is referenced by:  1stcof 4098  sincolem 8648

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-fun 3189  df-fn 3190  df-f 3191

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