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Theorem rngohomf 26596
Description: A ring homomorphism is a function. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
rnghomf.1  |-  G  =  ( 1st `  R
)
rnghomf.2  |-  X  =  ran  G
rnghomf.3  |-  J  =  ( 1st `  S
)
rnghomf.4  |-  Y  =  ran  J
Assertion
Ref Expression
rngohomf  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F : X
--> Y )

Proof of Theorem rngohomf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnghomf.1 . . . . 5  |-  G  =  ( 1st `  R
)
2 eqid 2438 . . . . 5  |-  ( 2nd `  R )  =  ( 2nd `  R )
3 rnghomf.2 . . . . 5  |-  X  =  ran  G
4 eqid 2438 . . . . 5  |-  (GId `  ( 2nd `  R ) )  =  (GId `  ( 2nd `  R ) )
5 rnghomf.3 . . . . 5  |-  J  =  ( 1st `  S
)
6 eqid 2438 . . . . 5  |-  ( 2nd `  S )  =  ( 2nd `  S )
7 rnghomf.4 . . . . 5  |-  Y  =  ran  J
8 eqid 2438 . . . . 5  |-  (GId `  ( 2nd `  S ) )  =  (GId `  ( 2nd `  S ) )
91, 2, 3, 4, 5, 6, 7, 8isrngohom 26595 . . . 4  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngHom  S )  <->  ( F : X --> Y  /\  ( F `  (GId `  ( 2nd `  R ) ) )  =  (GId `  ( 2nd `  S ) )  /\  A. x  e.  X  A. y  e.  X  ( ( F `  ( x G y ) )  =  ( ( F `
 x ) J ( F `  y
) )  /\  ( F `  ( x
( 2nd `  R
) y ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  y
) ) ) ) ) )
109biimpa 472 . . 3  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  -> 
( F : X --> Y  /\  ( F `  (GId `  ( 2nd `  R
) ) )  =  (GId `  ( 2nd `  S ) )  /\  A. x  e.  X  A. y  e.  X  (
( F `  (
x G y ) )  =  ( ( F `  x ) J ( F `  y ) )  /\  ( F `  ( x ( 2nd `  R
) y ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  y
) ) ) ) )
1110simp1d 970 . 2  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  ->  F : X --> Y )
12113impa 1149 1  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F : X
--> Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   ran crn 4882   -->wf 5453   ` cfv 5457  (class class class)co 6084   1stc1st 6350   2ndc2nd 6351  GIdcgi 21780   RingOpscrngo 21968    RngHom crnghom 26590
This theorem is referenced by:  rngohomcl  26597  rngogrphom  26601  rngohomco  26604  keridl  26656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-map 7023  df-rngohom 26593
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