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Theorem rngokerinj 26283
Description: A ring homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.)
Hypotheses
Ref Expression
rngkerinj.1  |-  G  =  ( 1st `  R
)
rngkerinj.2  |-  X  =  ran  G
rngkerinj.3  |-  W  =  (GId `  G )
rngkerinj.4  |-  J  =  ( 1st `  S
)
rngkerinj.5  |-  Y  =  ran  J
rngkerinj.6  |-  Z  =  (GId `  J )
Assertion
Ref Expression
rngokerinj  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F : X -1-1-> Y  <->  ( `' F " { Z } )  =  { W }
) )

Proof of Theorem rngokerinj
StepHypRef Expression
1 eqid 2388 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
21rngogrpo 21827 . . 3  |-  ( R  e.  RingOps  ->  ( 1st `  R
)  e.  GrpOp )
323ad2ant1 978 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( 1st `  R )  e.  GrpOp )
4 eqid 2388 . . . 4  |-  ( 1st `  S )  =  ( 1st `  S )
54rngogrpo 21827 . . 3  |-  ( S  e.  RingOps  ->  ( 1st `  S
)  e.  GrpOp )
653ad2ant2 979 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( 1st `  S )  e.  GrpOp )
71, 4rngogrphom 26279 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F  e.  ( ( 1st `  R
) GrpOpHom  ( 1st `  S
) ) )
8 rngkerinj.2 . . . 4  |-  X  =  ran  G
9 rngkerinj.1 . . . . 5  |-  G  =  ( 1st `  R
)
109rneqi 5037 . . . 4  |-  ran  G  =  ran  ( 1st `  R
)
118, 10eqtri 2408 . . 3  |-  X  =  ran  ( 1st `  R
)
12 rngkerinj.3 . . . 4  |-  W  =  (GId `  G )
139fveq2i 5672 . . . 4  |-  (GId `  G )  =  (GId
`  ( 1st `  R
) )
1412, 13eqtri 2408 . . 3  |-  W  =  (GId `  ( 1st `  R ) )
15 rngkerinj.5 . . . 4  |-  Y  =  ran  J
16 rngkerinj.4 . . . . 5  |-  J  =  ( 1st `  S
)
1716rneqi 5037 . . . 4  |-  ran  J  =  ran  ( 1st `  S
)
1815, 17eqtri 2408 . . 3  |-  Y  =  ran  ( 1st `  S
)
19 rngkerinj.6 . . . 4  |-  Z  =  (GId `  J )
2016fveq2i 5672 . . . 4  |-  (GId `  J )  =  (GId
`  ( 1st `  S
) )
2119, 20eqtri 2408 . . 3  |-  Z  =  (GId `  ( 1st `  S ) )
2211, 14, 18, 21grpokerinj 26252 . 2  |-  ( ( ( 1st `  R
)  e.  GrpOp  /\  ( 1st `  S )  e. 
GrpOp  /\  F  e.  ( ( 1st `  R
) GrpOpHom  ( 1st `  S
) ) )  -> 
( F : X -1-1-> Y  <-> 
( `' F " { Z } )  =  { W } ) )
233, 6, 7, 22syl3anc 1184 1  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F : X -1-1-> Y  <->  ( `' F " { Z } )  =  { W }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1649    e. wcel 1717   {csn 3758   `'ccnv 4818   ran crn 4820   "cima 4822   -1-1->wf1 5392   ` cfv 5395  (class class class)co 6021   1stc1st 6287   GrpOpcgr 21623  GIdcgi 21624   GrpOpHom cghom 21794   RingOpscrngo 21812    RngHom crnghom 26268
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-map 6957  df-grpo 21628  df-gid 21629  df-ginv 21630  df-gdiv 21631  df-ablo 21719  df-ghom 21795  df-rngo 21813  df-rngohom 26271
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