Theorem f1co 6110

Description: Composition of one-to-one functions. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by NM, 28-May-1998.)

Assertion

Ref	Expression
f1co	|- ( ( F : B -1-1-> C /\ G : A -1-1-> B ) -> ( F o. G ) : A -1-1-> C )

Proof of Theorem f1co

Step	Hyp	Ref	Expression
1		df-f1 5893	. . 3 |- ( F : B -1-1-> C <-> ( F : B --> C /\ Fun `' F ) )
2		df-f1 5893	. . 3 |- ( G : A -1-1-> B <-> ( G : A --> B /\ Fun `' G ) )
3		fco 6058	. . . . 5 |- ( ( F : B --> C /\ G : A --> B ) -> ( F o. G ) : A --> C )
4		funco 5928	. . . . . . 7 |- ( ( Fun `' G /\ Fun `' F ) -> Fun ( `' G o. `' F ) )
5		cnvco 5308	. . . . . . . 8 |- `' ( F o. G ) = ( `' G o. `' F )
6	5	funeqi 5909	. . . . . . 7 |- ( Fun `' ( F o. G ) <-> Fun ( `' G o. `' F ) )
7	4, 6	sylibr 224	. . . . . 6 |- ( ( Fun `' G /\ Fun `' F ) -> Fun `' ( F o. G ) )
8	7	ancoms 469	. . . . 5 |- ( ( Fun `' F /\ Fun `' G ) -> Fun `' ( F o. G ) )
9	3, 8	anim12i 590	. . . 4 |- ( ( ( F : B --> C /\ G : A --> B ) /\ ( Fun `' F /\ Fun `' G ) ) -> ( ( F o. G ) : A --> C /\ Fun `' ( F o. G ) ) )
10	9	an4s 869	. . 3 |- ( ( ( F : B --> C /\ Fun `' F ) /\ ( G : A --> B /\ Fun `' G ) ) -> ( ( F o. G ) : A --> C /\ Fun `' ( F o. G ) ) )
11	1, 2, 10	syl2anb 496	. 2 |- ( ( F : B -1-1-> C /\ G : A -1-1-> B ) -> ( ( F o. G ) : A --> C /\ Fun `' ( F o. G ) ) )
12		df-f1 5893	. 2 |- ( ( F o. G ) : A -1-1-> C <-> ( ( F o. G ) : A --> C /\ Fun `' ( F o. G ) ) )
13	11, 12	sylibr 224	1 |- ( ( F : B -1-1-> C /\ G : A -1-1-> B ) -> ( F o. G ) : A -1-1-> C )

Syntax hints:   -> wi 4   /\ wa 384   `'ccnv 5113   o. ccom 5118   Fun wfun 5882   -->wf 5884   -1-1->wf1 5885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893

This theorem is referenced by:  f1oco  6159  f1cofveqaeqALT  6516  tposf12  7377  domtr  8009  dfac12lem2  8966  fin23lem28  9162  pwfseqlem5  9485  cofth  16595  gsumzf1o  18313  erdsze2lem2  31186