Theorem funco 3536

Description: The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25.

Assertion

Ref	Expression
funco	|- ((Fun F /\ Fun G) -> Fun (F o. G))

Proof of Theorem funco

Step	Hyp	Ref	Expression
1		moexexv 1432	. . . . . . 7 |- ((E*z xGz /\ A.zE*y zFy) -> E*yE.z(xGz /\ zFy))
2		funmo 3518	. . . . . . 7 |- (Fun G -> E*z xGz)
3		dffunmo 3517	. . . . . . . 8 |- (Fun F <-> (Rel F /\ A.zE*y zFy))
4	3	pm3.27bi 326	. . . . . . 7 |- (Fun F -> A.zE*y zFy)
5	1, 2, 4	syl2an 454	. . . . . 6 |- ((Fun G /\ Fun F) -> E*yE.z(xGz /\ zFy))
6	5	ancoms 436	. . . . 5 |- ((Fun F /\ Fun G) -> E*yE.z(xGz /\ zFy))
7		visset 1804	. . . . . . 7 |- x e. V
8		visset 1804	. . . . . . 7 |- y e. V
9	7, 8	brco 3278	. . . . . 6 |- (x(F o. G)y <-> E.z(xGz /\ zFy))
10	9	mobii 1398	. . . . 5 |- (E*y x(F o. G)y <-> E*yE.z(xGz /\ zFy))
11	6, 10	sylibr 200	. . . 4 |- ((Fun F /\ Fun G) -> E*y x(F o. G)y)
12	11	19.21aiv 1281	. . 3 |- ((Fun F /\ Fun G) -> A.xE*y x(F o. G)y)
13		relco 3470	. . 3 |- Rel (F o. G)
14	12, 13	jctil 292	. 2 |- ((Fun F /\ Fun G) -> (Rel (F o. G) /\ A.xE*y x(F o. G)y))
15		dffunmo 3517	. 2 |- (Fun (F o. G) <-> (Rel (F o. G) /\ A.xE*y x(F o. G)y))
16	14, 15	sylibr 200	1 |- ((Fun F /\ Fun G) -> Fun (F o. G))

Syntax hints:   -> wi 3   /\ wa 223  A.wal 951  E.wex 977  E*wmo 1374   class class class wbr 2609   o. ccom 3164  Rel wrel 3165  Fun wfun 3166

This theorem is referenced by:  fnco 3581  fco 3621  f1co 3652  fvco 3759  curry1 4082  mapenlem1 4469  vsfval 8194

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-fun 3182

Copyright terms: Public domain