Theorem fnfco 3648

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 3601	. . 3 |- ((F Fn A /\ G Fn B /\ ran G (_ A) -> (F o. G) Fn B)
2	1	3expb 836	. 2 |- ((F Fn A /\ (G Fn B /\ ran G (_ A)) -> (F o. G) Fn B)
3		df-f 3200	. 2 |- (G:B-->A <-> (G Fn B /\ ran G (_ A))
4	2, 3	sylan2b 454	1 |- ((F Fn A /\ G:B-->A) -> (F o. G) Fn B)

Syntax hints:   -> wi 3   /\ wa 223   (_ wss 2050  ran crn 3177   o. ccom 3180   Fn wfn 3183  -->wf 3184

This theorem is referenced by:  1stcof 4107  sincolem 8660

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-fun 3198  df-fn 3199  df-f 3200

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