Theorem fcof1o 5768

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 5762	. . . 4 |- ( ( F : A --> B /\ ( G o. F ) = ( _I |` A ) ) -> F : A -1-1-> B )
2	1	ad2ant2rl 508	. . 3 |- ( ( ( F : A --> B /\ G : B --> A ) /\ ( ( F o. G ) = ( _I |` B ) /\ ( G o. F ) = ( _I |` A ) ) ) -> F : A -1-1-> B )
3		fcofo 5763	. . . . 5 |- ( ( F : A --> B /\ G : B --> A /\ ( F o. G ) = ( _I |` B ) ) -> F : A -onto-> B )
4	3	3expa 1198	. . . 4 |- ( ( ( F : A --> B /\ G : B --> A ) /\ ( F o. G ) = ( _I |` B ) ) -> F : A -onto-> B )
5	4	adantrr 476	. . 3 |- ( ( ( F : A --> B /\ G : B --> A ) /\ ( ( F o. G ) = ( _I |` B ) /\ ( G o. F ) = ( _I |` A ) ) ) -> F : A -onto-> B )
6		df-f1o 5205	. . 3 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
7	2, 5, 6	sylanbrc 415	. 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 526	. . . 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 4773	. . 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 5129	. . . 4 |- ( ( `' F o. F ) o. G ) = ( `' F o. ( F o. G ) )
11		f1ococnv1 5471	. . . . . . 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 4772	. . . . 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 5379	. . . . . 6 |- ( G : B --> A -> ( ( _I |` A ) o. G ) = G )
15	14	ad2antlr 486	. . . . 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 2203	. . . 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 2217	. . 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 5455	. . . 4 |- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A )
19		f1of 5442	. . . 4 |- ( `' F : B -1-1-onto-> A -> `' F : B --> A )
20		fcoi1 5378	. . . 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 2212	. 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 1348   _I cid 4273   `'ccnv 4610   |` cres 4613   o. ccom 4615   -->wf 5194   -1-1->wf1 5195   -onto->wfo 5196   -1-1-onto->wf1o 5197

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206

This theorem is referenced by:  txswaphmeo  13115