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Theorem injrec2 25222
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 5453 . . . . . . 7  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 fex 5765 . . . . . . 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 5224 . . . . . 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 5276 . . . . . . . 8  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
76simprbi 450 . . . . . . 7  |-  ( F : A -1-1-> B  ->  Fun  `' F )
8 f1cocnv1 5519 . . . . . . 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 5290 . . . . . . 7  |-  ( r  =  `' F  -> 
( Fun  r  <->  Fun  `' F
) )
12 coeq1 4857 . . . . . . . 8  |-  ( r  =  `' F  -> 
( r  o.  F
)  =  ( `' F  o.  F ) )
1312eqeq1d 2304 . . . . . . 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 2882 . . . . 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 5813 . . . . . 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 1626 . . 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 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801    _I cid 4320   `'ccnv 4704    |` cres 4707    o. ccom 4709   Fun wfun 5265   -->wf 5267   -1-1->wf1 5268
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279
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