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Theorem subgornss 25286
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 25284 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  =  ( G  |`  ( W  X.  W ) ) )
32rneqd 5220 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ran  H  =  ran  ( G  |`  ( W  X.  W
) ) )
4 df-ima 5002 . . . 4  |-  ( G
" ( W  X.  W ) )  =  ran  ( G  |`  ( W  X.  W
) )
53, 4syl6eqr 2502 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ran  H  =  ( G " ( W  X.  W ) ) )
6 imassrn 5338 . . 3  |-  ( G
" ( W  X.  W ) )  C_  ran  G
75, 6syl6eqss 3539 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ran  H  C_  ran  G )
8 subgornss.1 . 2  |-  X  =  ran  G
97, 1, 83sstr4g 3530 1  |-  ( H  e.  ( SubGrpOp `  G
)  ->  W  C_  X
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804    C_ wss 3461    X. cxp 4987   ran crn 4990    |` cres 4991   "cima 4992   ` cfv 5578   SubGrpOpcsubgo 25281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fo 5584  df-fv 5586  df-ov 6284  df-grpo 25171  df-subgo 25282
This theorem is referenced by:  subgoid  25287  subgoinv  25288  subgoablo  25291  ghsubgolemOLD  25350
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