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Theorem subgornss 20973
Description: The underlying set of a subgroup is a subset of its parent group's underlying set. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
subgornss.1  |-  X  =  ran  G
subgornss.2  |-  W  =  ran  H
Assertion
Ref Expression
subgornss  |-  ( H  e.  ( SubGrpOp `  G
)  ->  W  C_  X
)

Proof of Theorem subgornss
StepHypRef Expression
1 imassrn 5025 . . 3  |-  ( G
" ( W  X.  W ) )  C_  ran  G
2 subgornss.2 . . . . . . 7  |-  W  =  ran  H
32subgores 20971 . . . . . 6  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  =  ( G  |`  ( W  X.  W ) ) )
43rneqd 4906 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ran  H  =  ran  ( G  |`  ( W  X.  W
) ) )
5 df-ima 4702 . . . . 5  |-  ( G
" ( W  X.  W ) )  =  ran  ( G  |`  ( W  X.  W
) )
64, 5syl6eqr 2333 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ran  H  =  ( G " ( W  X.  W ) ) )
76sseq1d 3205 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( ran  H 
C_  ran  G  <->  ( G " ( W  X.  W
) )  C_  ran  G ) )
81, 7mpbiri 224 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ran  H  C_  ran  G )
9 subgornss.1 . 2  |-  X  =  ran  G
108, 2, 93sstr4g 3219 1  |-  ( H  e.  ( SubGrpOp `  G
)  ->  W  C_  X
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    C_ wss 3152    X. cxp 4687   ran crn 4690    |` cres 4691   "cima 4692   ` cfv 5255   SubGrpOpcsubgo 20968
This theorem is referenced by:  subgoid  20974  subgoinv  20975  subgoablo  20978  ghsubgolem  21037
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-csb 3082  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-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-grpo 20858  df-subgo 20969
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