Theorem fcof1o 5964

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 5958	. . . 4 |- ( ( F : A --> B /\ ( G o. F ) = ( _I |` A ) ) -> F : A -1-1-> B )
2	1	ad2ant2rl 511	. . 3 |- ( ( ( F : A --> B /\ G : B --> A ) /\ ( ( F o. G ) = ( _I |` B ) /\ ( G o. F ) = ( _I |` A ) ) ) -> F : A -1-1-> B )
3		fcofo 5959	. . . . 5 |- ( ( F : A --> B /\ G : B --> A /\ ( F o. G ) = ( _I |` B ) ) -> F : A -onto-> B )
4	3	3expa 1230	. . . 4 |- ( ( ( F : A --> B /\ G : B --> A ) /\ ( F o. G ) = ( _I |` B ) ) -> F : A -onto-> B )
5	4	adantrr 479	. . 3 |- ( ( ( F : A --> B /\ G : B --> A ) /\ ( ( F o. G ) = ( _I |` B ) /\ ( G o. F ) = ( _I |` A ) ) ) -> F : A -onto-> B )
6		df-f1o 5361	. . 3 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
7	2, 5, 6	sylanbrc 417	. 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 531	. . . 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 4919	. . 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 5283	. . . 4 |- ( ( `' F o. F ) o. G ) = ( `' F o. ( F o. G ) )
11		f1ococnv1 5645	. . . . . . 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 4918	. . . . 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 5550	. . . . . 6 |- ( G : B --> A -> ( ( _I |` A ) o. G ) = G )
15	14	ad2antlr 489	. . . . 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 2267	. . . 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 2281	. . 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 5629	. . . 4 |- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A )
19		f1of 5616	. . . 4 |- ( `' F : B -1-1-onto-> A -> `' F : B --> A )
20		fcoi1 5549	. . . 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 2276	. 2 |- ( ( ( F : A --> B /\ G : B --> A ) /\ ( ( F o. G ) = ( _I |` B ) /\ ( G o. F ) = ( _I |` A ) ) ) -> `' F = G )
23	7, 22	jca 306	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 104   = wceq 1398   _I cid 4411   `'ccnv 4750   |` cres 4753   o. ccom 4755   -->wf 5350   -1-1->wf1 5351   -onto->wfo 5352   -1-1-onto->wf1o 5353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362

This theorem is referenced by:  txswaphmeo  15203