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Theorem fcof1o 6188
Description: Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.) (Proof shortened by AV, 15-Dec-2019.)
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
StepHypRef Expression
1 simpll 753 . . 3  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  F : A --> B )
2 simpr 461 . . . 4  |-  ( ( F : A --> B  /\  G : B --> A )  ->  G : B --> A )
32adantr 465 . . 3  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  G : B --> A )
4 simpr 461 . . . 4  |-  ( ( ( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) )  ->  ( G  o.  F )  =  (  _I  |`  A )
)
54adantl 466 . . 3  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  ( G  o.  F )  =  (  _I  |`  A ) )
6 simpl 457 . . . 4  |-  ( ( ( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) )  ->  ( F  o.  G )  =  (  _I  |`  B )
)
76adantl 466 . . 3  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  ( F  o.  G )  =  (  _I  |`  B ) )
81, 3, 5, 7fcof1od 6186 . 2  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  F : A -1-1-onto-> B )
9 simpl 457 . . . 4  |-  ( ( F : A --> B  /\  G : B --> A )  ->  F : A --> B )
109adantr 465 . . 3  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  F : A --> B )
1110, 3, 5, 72fcoidinvd 6187 . 2  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  `' F  =  G )
128, 11jca 532 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 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    _I cid 4790   `'ccnv 4998    |` cres 5001    o. ccom 5003   -->wf 5584   -1-1-onto->wf1o 5587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596
This theorem is referenced by:  setcinv  15278  catciso  15295  yonedainv  15411  pmtrff1o  16303  pmtrfcnv  16304  evpmodpmf1o  18439  txswaphmeo  20133  qtophmeo  20145
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