Theorem funco 3547

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 1439	. . . . . . 7 |- ((E*z xGz /\ A.zE*y zFy) -> E*yE.z(xGz /\ zFy))
2		funmo 3529	. . . . . . 7 |- (Fun G -> E*z xGz)
3		dffunmo 3528	. . . . . . . 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 1811	. . . . . . 7 |- x e. V
8		visset 1811	. . . . . . 7 |- y e. V
9	7, 8	brco 3286	. . . . . 6 |- (x(F o. G)y <-> E.z(xGz /\ zFy))
10	9	mobii 1405	. . . . 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 1286	. . 3 |- ((Fun F /\ Fun G) -> A.xE*y x(F o. G)y)
13		relco 3481	. . 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 3528	. 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 953  E.wex 979  E*wmo 1381   class class class wbr 2616   o. ccom 3171  Rel wrel 3172  Fun wfun 3173

This theorem is referenced by:  fnco 3592  fco 3633  f1co 3664  fvco 3771  curry1 4095  mapenlem1 4482  vsfval 8239

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-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-fun 3189

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