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Theorem subgornss 21742
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 subgornss.2 . . . . . 6  |-  W  =  ran  H
21subgores 21740 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  =  ( G  |`  ( W  X.  W ) ) )
32rneqd 5037 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ran  H  =  ran  ( G  |`  ( W  X.  W
) ) )
4 df-ima 4831 . . . 4  |-  ( G
" ( W  X.  W ) )  =  ran  ( G  |`  ( W  X.  W
) )
53, 4syl6eqr 2437 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ran  H  =  ( G " ( W  X.  W ) ) )
6 imassrn 5156 . . 3  |-  ( G
" ( W  X.  W ) )  C_  ran  G
75, 6syl6eqss 3341 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ran  H  C_  ran  G )
8 subgornss.1 . 2  |-  X  =  ran  G
97, 1, 83sstr4g 3332 1  |-  ( H  e.  ( SubGrpOp `  G
)  ->  W  C_  X
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    C_ wss 3263    X. cxp 4816   ran crn 4819    |` cres 4820   "cima 4821   ` cfv 5394   SubGrpOpcsubgo 21737
This theorem is referenced by:  subgoid  21743  subgoinv  21744  subgoablo  21747  ghsubgolem  21806
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 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fo 5400  df-fv 5402  df-ov 6023  df-grpo 21627  df-subgo 21738
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