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Theorem rngohomcl 26584
Description: Closure law for a ring homomorphism. (Contributed by Jeff Madsen, 3-Jan-2011.)
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
rngohomcl  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  A  e.  X )  ->  ( F `  A )  e.  Y )

Proof of Theorem rngohomcl
StepHypRef Expression
1 rnghomf.1 . . 3  |-  G  =  ( 1st `  R
)
2 rnghomf.2 . . 3  |-  X  =  ran  G
3 rnghomf.3 . . 3  |-  J  =  ( 1st `  S
)
4 rnghomf.4 . . 3  |-  Y  =  ran  J
51, 2, 3, 4rngohomf 26583 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F : X
--> Y )
65ffvelrnda 5871 1  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  A  e.  X )  ->  ( F `  A )  e.  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ran crn 4880   ` cfv 5455  (class class class)co 6082   1stc1st 6348   RingOpscrngo 21964    RngHom crnghom 26577
This theorem is referenced by:  rngohomco  26591  keridl  26643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-map 7021  df-rngohom 26580
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