Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  injrec2 Unicode version

Theorem injrec2 25119
Description: A function is an injection iff a retraction exists. Bourbaki E.II.18 prop. 8. (Contributed by FL, 11-Nov-2011.)
Assertion
Ref Expression
injrec2  |-  ( ( F : A --> B  /\  A  e.  C )  ->  ( F : A -1-1-> B  <->  E. r ( Fun  r  /\  ( r  o.  F
)  =  (  _I  |`  A ) ) ) )
Distinct variable groups:    A, r    B, r    F, r
Allowed substitution hint:    C( r)

Proof of Theorem injrec2
StepHypRef Expression
1 f1f 5437 . . . . . . 7  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 fex 5749 . . . . . . 7  |-  ( ( F : A --> B  /\  A  e.  C )  ->  F  e.  _V )
31, 2sylan 457 . . . . . 6  |-  ( ( F : A -1-1-> B  /\  A  e.  C
)  ->  F  e.  _V )
4 cnvexg 5208 . . . . . 6  |-  ( F  e.  _V  ->  `' F  e.  _V )
53, 4syl 15 . . . . 5  |-  ( ( F : A -1-1-> B  /\  A  e.  C
)  ->  `' F  e.  _V )
6 df-f1 5260 . . . . . . . 8  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
76simprbi 450 . . . . . . 7  |-  ( F : A -1-1-> B  ->  Fun  `' F )
8 f1cocnv1 5503 . . . . . . 7  |-  ( F : A -1-1-> B  -> 
( `' F  o.  F )  =  (  _I  |`  A )
)
97, 8jca 518 . . . . . 6  |-  ( F : A -1-1-> B  -> 
( Fun  `' F  /\  ( `' F  o.  F )  =  (  _I  |`  A )
) )
109adantr 451 . . . . 5  |-  ( ( F : A -1-1-> B  /\  A  e.  C
)  ->  ( Fun  `' F  /\  ( `' F  o.  F )  =  (  _I  |`  A ) ) )
11 funeq 5274 . . . . . . 7  |-  ( r  =  `' F  -> 
( Fun  r  <->  Fun  `' F
) )
12 coeq1 4841 . . . . . . . 8  |-  ( r  =  `' F  -> 
( r  o.  F
)  =  ( `' F  o.  F ) )
1312eqeq1d 2291 . . . . . . 7  |-  ( r  =  `' F  -> 
( ( r  o.  F )  =  (  _I  |`  A )  <->  ( `' F  o.  F
)  =  (  _I  |`  A ) ) )
1411, 13anbi12d 691 . . . . . 6  |-  ( r  =  `' F  -> 
( ( Fun  r  /\  ( r  o.  F
)  =  (  _I  |`  A ) )  <->  ( Fun  `' F  /\  ( `' F  o.  F )  =  (  _I  |`  A ) ) ) )
1514spcegv 2869 . . . . 5  |-  ( `' F  e.  _V  ->  ( ( Fun  `' F  /\  ( `' F  o.  F )  =  (  _I  |`  A )
)  ->  E. r
( Fun  r  /\  ( r  o.  F
)  =  (  _I  |`  A ) ) ) )
165, 10, 15sylc 56 . . . 4  |-  ( ( F : A -1-1-> B  /\  A  e.  C
)  ->  E. r
( Fun  r  /\  ( r  o.  F
)  =  (  _I  |`  A ) ) )
1716expcom 424 . . 3  |-  ( A  e.  C  ->  ( F : A -1-1-> B  ->  E. r ( Fun  r  /\  ( r  o.  F
)  =  (  _I  |`  A ) ) ) )
1817adantl 452 . 2  |-  ( ( F : A --> B  /\  A  e.  C )  ->  ( F : A -1-1-> B  ->  E. r ( Fun  r  /\  ( r  o.  F )  =  (  _I  |`  A ) ) ) )
19 fcof1 5797 . . . . . 6  |-  ( ( F : A --> B  /\  ( r  o.  F
)  =  (  _I  |`  A ) )  ->  F : A -1-1-> B )
2019ex 423 . . . . 5  |-  ( F : A --> B  -> 
( ( r  o.  F )  =  (  _I  |`  A )  ->  F : A -1-1-> B
) )
2120adantld 453 . . . 4  |-  ( F : A --> B  -> 
( ( Fun  r  /\  ( r  o.  F
)  =  (  _I  |`  A ) )  ->  F : A -1-1-> B ) )
2221exlimdv 1664 . . 3  |-  ( F : A --> B  -> 
( E. r ( Fun  r  /\  (
r  o.  F )  =  (  _I  |`  A ) )  ->  F : A -1-1-> B ) )
2322adantr 451 . 2  |-  ( ( F : A --> B  /\  A  e.  C )  ->  ( E. r ( Fun  r  /\  (
r  o.  F )  =  (  _I  |`  A ) )  ->  F : A -1-1-> B ) )
2418, 23impbid 183 1  |-  ( ( F : A --> B  /\  A  e.  C )  ->  ( F : A -1-1-> B  <->  E. r ( Fun  r  /\  ( r  o.  F
)  =  (  _I  |`  A ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    _I cid 4304   `'ccnv 4688    |` cres 4691    o. ccom 4693   Fun wfun 5249   -->wf 5251   -1-1->wf1 5252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
  Copyright terms: Public domain W3C validator