Theorem fcof1o 5757

Description: Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
fcof1o	|- ( ( ( F : A --> B /\ G : B --> A ) /\ ( ( F o. G ) = ( _I |` B ) /\ ( G o. F ) = ( _I |` A ) ) ) -> ( F : A -1-1-onto-> B /\ `' F = G ) )

Proof of Theorem fcof1o

Step	Hyp	Ref	Expression
1		fcof1 5751	. . . 4 |- ( ( F : A --> B /\ ( G o. F ) = ( _I |` A ) ) -> F : A -1-1-> B )
2	1	ad2ant2rl 503	. . 3 |- ( ( ( F : A --> B /\ G : B --> A ) /\ ( ( F o. G ) = ( _I |` B ) /\ ( G o. F ) = ( _I |` A ) ) ) -> F : A -1-1-> B )
3		fcofo 5752	. . . . 5 |- ( ( F : A --> B /\ G : B --> A /\ ( F o. G ) = ( _I |` B ) ) -> F : A -onto-> B )
4	3	3expa 1193	. . . 4 |- ( ( ( F : A --> B /\ G : B --> A ) /\ ( F o. G ) = ( _I |` B ) ) -> F : A -onto-> B )
5	4	adantrr 471	. . 3 |- ( ( ( F : A --> B /\ G : B --> A ) /\ ( ( F o. G ) = ( _I |` B ) /\ ( G o. F ) = ( _I |` A ) ) ) -> F : A -onto-> B )
6		df-f1o 5195	. . 3 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
7	2, 5, 6	sylanbrc 414	. 2 |- ( ( ( F : A --> B /\ G : B --> A ) /\ ( ( F o. G ) = ( _I |` B ) /\ ( G o. F ) = ( _I |` A ) ) ) -> F : A -1-1-onto-> B )
8		simprl 521	. . . 4 |- ( ( ( F : A --> B /\ G : B --> A ) /\ ( ( F o. G ) = ( _I |` B ) /\ ( G o. F ) = ( _I |` A ) ) ) -> ( F o. G ) = ( _I |` B ) )
9	8	coeq2d 4766	. . 3 |- ( ( ( F : A --> B /\ G : B --> A ) /\ ( ( F o. G ) = ( _I |` B ) /\ ( G o. F ) = ( _I |` A ) ) ) -> ( `' F o. ( F o. G ) ) = ( `' F o. ( _I |` B ) ) )
10		coass 5122	. . . 4 |- ( ( `' F o. F ) o. G ) = ( `' F o. ( F o. G ) )
11		f1ococnv1 5461	. . . . . . 7 |- ( F : A -1-1-onto-> B -> ( `' F o. F ) = ( _I |` A ) )
12	7, 11	syl 14	. . . . . 6 |- ( ( ( F : A --> B /\ G : B --> A ) /\ ( ( F o. G ) = ( _I |` B ) /\ ( G o. F ) = ( _I |` A ) ) ) -> ( `' F o. F ) = ( _I |` A ) )
13	12	coeq1d 4765	. . . . 5 |- ( ( ( F : A --> B /\ G : B --> A ) /\ ( ( F o. G ) = ( _I |` B ) /\ ( G o. F ) = ( _I |` A ) ) ) -> ( ( `' F o. F ) o. G ) = ( ( _I |` A ) o. G ) )
14		fcoi2 5369	. . . . . 6 |- ( G : B --> A -> ( ( _I |` A ) o. G ) = G )
15	14	ad2antlr 481	. . . . 5 |- ( ( ( F : A --> B /\ G : B --> A ) /\ ( ( F o. G ) = ( _I |` B ) /\ ( G o. F ) = ( _I |` A ) ) ) -> ( ( _I |` A ) o. G ) = G )
16	13, 15	eqtrd 2198	. . . 4 |- ( ( ( F : A --> B /\ G : B --> A ) /\ ( ( F o. G ) = ( _I |` B ) /\ ( G o. F ) = ( _I |` A ) ) ) -> ( ( `' F o. F ) o. G ) = G )
17	10, 16	eqtr3id 2213	. . 3 |- ( ( ( F : A --> B /\ G : B --> A ) /\ ( ( F o. G ) = ( _I |` B ) /\ ( G o. F ) = ( _I |` A ) ) ) -> ( `' F o. ( F o. G ) ) = G )
18		f1ocnv 5445	. . . 4 |- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A )
19		f1of 5432	. . . 4 |- ( `' F : B -1-1-onto-> A -> `' F : B --> A )
20		fcoi1 5368	. . . 4 |- ( `' F : B --> A -> ( `' F o. ( _I |` B ) ) = `' F )
21	7, 18, 19, 20	4syl 18	. . 3 |- ( ( ( F : A --> B /\ G : B --> A ) /\ ( ( F o. G ) = ( _I |` B ) /\ ( G o. F ) = ( _I |` A ) ) ) -> ( `' F o. ( _I |` B ) ) = `' F )
22	9, 17, 21	3eqtr3rd 2207	. 2 |- ( ( ( F : A --> B /\ G : B --> A ) /\ ( ( F o. G ) = ( _I |` B ) /\ ( G o. F ) = ( _I |` A ) ) ) -> `' F = G )
23	7, 22	jca 304	1 |- ( ( ( F : A --> B /\ G : B --> A ) /\ ( ( F o. G ) = ( _I |` B ) /\ ( G o. F ) = ( _I |` A ) ) ) -> ( F : A -1-1-onto-> B /\ `' F = G ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   _I cid 4266   `'ccnv 4603   |` cres 4606   o. ccom 4608   -->wf 5184   -1-1->wf1 5185   -onto->wfo 5186   -1-1-onto->wf1o 5187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196

This theorem is referenced by:  txswaphmeo  12961