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Theorem subgores 21894
 Description: A subgroup operation is the restriction of its parent group operation to its underlying set. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
subgores.1
Assertion
Ref Expression
subgores

Proof of Theorem subgores
StepHypRef Expression
1 issubgo 21893 . . . . 5
21simp1bi 973 . . . 4
3 eqid 2438 . . . . 5
43grpofo 21789 . . . 4
5 fofun 5656 . . . 4
62, 4, 53syl 19 . . 3
71simp3bi 975 . . 3
81simp2bi 974 . . . . 5
9 subgores.1 . . . . . 6
109grpofo 21789 . . . . 5
11 fof 5655 . . . . 5
128, 10, 113syl 19 . . . 4
13 fdm 5597 . . . 4
14 eqimss2 3403 . . . 4
1512, 13, 143syl 19 . . 3
16 fun2ssres 5496 . . 3
176, 7, 15, 16syl3anc 1185 . 2
18 fofn 5657 . . 3
19 fnresdm 5556 . . 3
208, 10, 18, 194syl 20 . 2
2117, 20eqtr2d 2471 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1653   wcel 1726   wss 3322   cxp 4878   cdm 4880   crn 4881   cres 4882   wfun 5450   wfn 5451  wf 5452  wfo 5454  cfv 5456  cgr 21776  csubgo 21891 This theorem is referenced by:  subgoov  21895  subgornss  21896 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fo 5462  df-fv 5464  df-ov 6086  df-grpo 21781  df-subgo 21892
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