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Theorem rngohomcl 26598
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 26597 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F : X
--> Y )
6 ffvelrn 5663 . 2  |-  ( ( F : X --> Y  /\  A  e.  X )  ->  ( F `  A
)  e.  Y )
75, 6sylan 457 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   RingOpscrngo 21042    RngHom crnghom 26591
This theorem is referenced by:  rngohomco  26605  keridl  26657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-rngohom 26594
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