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Theorem grpoinvcl 28310
Description: A group element's inverse is a group element. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpinvcl.1 𝑋 = ran 𝐺
grpinvcl.2 𝑁 = (inv‘𝐺)
Assertion
Ref Expression
grpoinvcl ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)

Proof of Theorem grpoinvcl
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 grpinvcl.1 . . 3 𝑋 = ran 𝐺
2 eqid 2824 . . 3 (GId‘𝐺) = (GId‘𝐺)
3 grpinvcl.2 . . 3 𝑁 = (inv‘𝐺)
41, 2, 3grpoinvval 28309 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝑦𝑋 (𝑦𝐺𝐴) = (GId‘𝐺)))
51, 2grpoinveu 28305 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∃!𝑦𝑋 (𝑦𝐺𝐴) = (GId‘𝐺))
6 riotacl 7124 . . 3 (∃!𝑦𝑋 (𝑦𝐺𝐴) = (GId‘𝐺) → (𝑦𝑋 (𝑦𝐺𝐴) = (GId‘𝐺)) ∈ 𝑋)
75, 6syl 17 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑦𝑋 (𝑦𝐺𝐴) = (GId‘𝐺)) ∈ 𝑋)
84, 7eqeltrd 2916 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  ∃!wreu 3135  ran crn 5543  cfv 6343  crio 7106  (class class class)co 7149  GrpOpcgr 28275  GIdcgi 28276  invcgn 28277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-grpo 28279  df-gid 28280  df-ginv 28281
This theorem is referenced by:  grpoinvid1  28314  grpoinvid2  28315  grpolcan  28316  grpo2inv  28317  grpoinvf  28318  grpoinvop  28319  grpodivinv  28322  grpoinvdiv  28323  grpodivf  28324  grpomuldivass  28327  grponpcan  28329  ablodivdiv4  28340  vcm  28362  rngonegcl  35310  isdrngo2  35341
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