Theorem fcof1o 5780

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 5774	. . . 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 5775	. . . . 5 |- ( ( F : A --> B /\ G : B --> A /\ ( F o. G ) = ( _I |` B ) ) -> F : A -onto-> B )
4	3	3expa 1203	. . . 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 5215	. . 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 529	. . . 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 4782	. . 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 5139	. . . 4 |- ( ( `' F o. F ) o. G ) = ( `' F o. ( F o. G ) )
11		f1ococnv1 5482	. . . . . . 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 4781	. . . . 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 5389	. . . . . 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 2208	. . . 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 2222	. . 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 5466	. . . 4 |- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A )
19		f1of 5453	. . . 4 |- ( `' F : B -1-1-onto-> A -> `' F : B --> A )
20		fcoi1 5388	. . . 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 2217	. 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 1353   _I cid 4282   `'ccnv 4619   |` cres 4622   o. ccom 4624   -->wf 5204   -1-1->wf1 5205   -onto->wfo 5206   -1-1-onto->wf1o 5207

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216

This theorem is referenced by:  txswaphmeo  13401