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Theorem subgornss 25425
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 25423 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  =  ( G  |`  ( W  X.  W ) ) )
32rneqd 5143 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ran  H  =  ran  ( G  |`  ( W  X.  W
) ) )
4 df-ima 4926 . . . 4  |-  ( G
" ( W  X.  W ) )  =  ran  ( G  |`  ( W  X.  W
) )
53, 4syl6eqr 2441 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ran  H  =  ( G " ( W  X.  W ) ) )
6 imassrn 5260 . . 3  |-  ( G
" ( W  X.  W ) )  C_  ran  G
75, 6syl6eqss 3467 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ran  H  C_  ran  G )
8 subgornss.1 . 2  |-  X  =  ran  G
97, 1, 83sstr4g 3458 1  |-  ( H  e.  ( SubGrpOp `  G
)  ->  W  C_  X
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399    e. wcel 1826    C_ wss 3389    X. cxp 4911   ran crn 4914    |` cres 4915   "cima 4916   ` cfv 5496   SubGrpOpcsubgo 25420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-fo 5502  df-fv 5504  df-ov 6199  df-grpo 25310  df-subgo 25421
This theorem is referenced by:  subgoid  25426  subgoinv  25427  subgoablo  25430  ghsubgolemOLD  25489
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