Theorem fcof1od 6549

Description: A function is bijective if a "retraction" and a "section" exist, see comments for fcof1 6542 and fcofo 6543. Formerly part of proof of fcof1o 6551. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by AV, 15-Dec-2019.)

Hypotheses

Ref	Expression
fcof1od.f	|- ( ph -> F : A --> B )
fcof1od.g	|- ( ph -> G : B --> A )
fcof1od.a	|- ( ph -> ( G o. F ) = ( _I |` A ) )
fcof1od.b	|- ( ph -> ( F o. G ) = ( _I |` B ) )

Assertion

Ref	Expression
fcof1od	|- ( ph -> F : A -1-1-onto-> B )

Proof of Theorem fcof1od

Step	Hyp	Ref	Expression
1		fcof1od.f	. . 3 |- ( ph -> F : A --> B )
2		fcof1od.a	. . 3 |- ( ph -> ( G o. F ) = ( _I |` A ) )
3		fcof1 6542	. . 3 |- ( ( F : A --> B /\ ( G o. F ) = ( _I |` A ) ) -> F : A -1-1-> B )
4	1, 2, 3	syl2anc 693	. 2 |- ( ph -> F : A -1-1-> B )
5		fcof1od.g	. . 3 |- ( ph -> G : B --> A )
6		fcof1od.b	. . 3 |- ( ph -> ( F o. G ) = ( _I |` B ) )
7		fcofo 6543	. . 3 |- ( ( F : A --> B /\ G : B --> A /\ ( F o. G ) = ( _I |` B ) ) -> F : A -onto-> B )
8	1, 5, 6, 7	syl3anc 1326	. 2 |- ( ph -> F : A -onto-> B )
9		df-f1o 5895	. 2 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
10	4, 8, 9	sylanbrc 698	1 |- ( ph -> F : A -1-1-onto-> B )

Syntax hints:   -> wi 4   = wceq 1483   _I cid 5023   |` cres 5116   o. ccom 5118   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887

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-8 1992  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-pow 4843  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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  2fcoidinvd  6550  fcof1o  6551  2fvidf1od  6553  catciso  16757  pmtrff1o  17883  evpmodpmf1o  19942