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Theorem infemb 25962
Description: The inclusion functor is an embedding. (Contributed by FL, 2-Nov-2009.)
Hypotheses
Ref Expression
infemb.1  |-  M1  =  dom  ( dom_ `  T
)
infemb.2  |-  M 2  =  dom  ( dom_ `  U
)
Assertion
Ref Expression
infemb  |-  ( U  e.  (  SubCat  `  T
)  ->  (  _I  |`  M 2 ) : M 2 -1-1-> M1 )

Proof of Theorem infemb
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 f1oi 5527 . . 3  |-  (  _I  |`  M 2 ) : M 2 -1-1-onto-> M 2
2 infemb.1 . . . 4  |-  M1  =  dom  ( dom_ `  T
)
3 infemb.2 . . . 4  |-  M 2  =  dom  ( dom_ `  U
)
42, 3morsubc 25958 . . 3  |-  ( U  e.  (  SubCat  `  T
)  ->  M 2  C_  M1 )
5 fvex 5555 . . . . . 6  |-  ( dom_ `  U )  e.  _V
65dmex 4957 . . . . 5  |-  dom  ( dom_ `  U )  e. 
_V
73, 6eqeltri 2366 . . . 4  |-  M 2  e.  _V
8 f1oeq3 5481 . . . . 5  |-  ( x  =  M 2  ->  ( (  _I  |`  M 2
) : M 2 -1-1-onto-> x  <->  (  _I  |`  M 2
) : M 2 -1-1-onto-> M 2 ) )
9 sseq1 3212 . . . . 5  |-  ( x  =  M 2  ->  ( x  C_  M1  <->  M 2  C_  M1 ) )
108, 9anbi12d 691 . . . 4  |-  ( x  =  M 2  ->  ( ( (  _I  |`  M 2
) : M 2 -1-1-onto-> x  /\  x  C_  M1 )  <->  ( (  _I  |`  M 2
) : M 2 -1-1-onto-> M 2  /\  M 2  C_  M1 ) ) )
117, 10spcev 2888 . . 3  |-  ( ( (  _I  |`  M 2
) : M 2 -1-1-onto-> M 2  /\  M 2  C_  M1 )  ->  E. x
( (  _I  |`  M 2
) : M 2 -1-1-onto-> x  /\  x  C_  M1 )
)
121, 4, 11sylancr 644 . 2  |-  ( U  e.  (  SubCat  `  T
)  ->  E. x
( (  _I  |`  M 2
) : M 2 -1-1-onto-> x  /\  x  C_  M1 )
)
13 resiexg 5013 . . . 4  |-  ( M 2  e.  _V  ->  (  _I  |`  M 2
)  e.  _V )
147, 13ax-mp 8 . . 3  |-  (  _I  |`  M 2 )  e. 
_V
1514f11o 5522 . 2  |-  ( (  _I  |`  M 2
) : M 2 -1-1-> M1 
<->  E. x ( (  _I  |`  M 2
) : M 2 -1-1-onto-> x  /\  x  C_  M1 )
)
1612, 15sylibr 203 1  |-  ( U  e.  (  SubCat  `  T
)  ->  (  _I  |`  M 2 ) : M 2 -1-1-> M1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165    _I cid 4320   dom cdm 4705    |` cres 4707   -1-1->wf1 5268   -1-1-onto->wf1o 5270   ` cfv 5271   dom_cdom_ 25815    SubCat csubcat 25946
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-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-rab 2565  df-v 2803  df-sbc 3005  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-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  df-1st 6138  df-2nd 6139  df-dom_ 25820  df-cod_ 25821  df-id_ 25822  df-cmpa 25823  df-catOLD 25856  df-subcat 25947
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