Theorem fcof1o 5988

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 5982	. . . 4 |- ( ( F : A --> B /\ ( G o. F ) = ( _I |` A ) ) -> F : A -1-1-> B )
2	1	ad2ant2rl 515	. . 3 |- ( ( ( F : A --> B /\ G : B --> A ) /\ ( ( F o. G ) = ( _I |` B ) /\ ( G o. F ) = ( _I |` A ) ) ) -> F : A -1-1-> B )
3		fcofo 5983	. . . . 5 |- ( ( F : A --> B /\ G : B --> A /\ ( F o. G ) = ( _I |` B ) ) -> F : A -onto-> B )
4	3	3expa 1234	. . . 4 |- ( ( ( F : A --> B /\ G : B --> A ) /\ ( F o. G ) = ( _I |` B ) ) -> F : A -onto-> B )
5	4	adantrr 483	. . 3 |- ( ( ( F : A --> B /\ G : B --> A ) /\ ( ( F o. G ) = ( _I |` B ) /\ ( G o. F ) = ( _I |` A ) ) ) -> F : A -onto-> B )
6		df-f1o 5382	. . 3 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
7	2, 5, 6	sylanbrc 421	. 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 535	. . . 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 4940	. . 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 5304	. . . 4 |- ( ( `' F o. F ) o. G ) = ( `' F o. ( F o. G ) )
11		f1ococnv1 5666	. . . . . . 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 4939	. . . . 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 5571	. . . . . 6 |- ( G : B --> A -> ( ( _I |` A ) o. G ) = G )
15	14	ad2antlr 493	. . . . 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 2271	. . . 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 2285	. . 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 5650	. . . 4 |- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A )
19		f1of 5637	. . . 4 |- ( `' F : B -1-1-onto-> A -> `' F : B --> A )
20		fcoi1 5570	. . . 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 2280	. 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 1402   _I cid 4431   `'ccnv 4771   |` cres 4774   o. ccom 4776   -->wf 5371   -1-1->wf1 5372   -onto->wfo 5373   -1-1-onto->wf1o 5374

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383

This theorem is referenced by:  txswaphmeo  15348