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Theorem grpoinvcl 29508
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 = (invβ€˜πΊ)
Assertion
Ref Expression
grpoinvcl ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) ∈ 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 2733 . . 3 (GIdβ€˜πΊ) = (GIdβ€˜πΊ)
3 grpinvcl.2 . . 3 𝑁 = (invβ€˜πΊ)
41, 2, 3grpoinvval 29507 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GIdβ€˜πΊ)))
51, 2grpoinveu 29503 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ βˆƒ!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GIdβ€˜πΊ))
6 riotacl 7332 . . 3 (βˆƒ!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GIdβ€˜πΊ) β†’ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GIdβ€˜πΊ)) ∈ 𝑋)
75, 6syl 17 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GIdβ€˜πΊ)) ∈ 𝑋)
84, 7eqeltrd 2834 1 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆƒ!wreu 3350  ran crn 5635  β€˜cfv 6497  β„©crio 7313  (class class class)co 7358  GrpOpcgr 29473  GIdcgi 29474  invcgn 29475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-grpo 29477  df-gid 29478  df-ginv 29479
This theorem is referenced by:  grpoinvid1  29512  grpoinvid2  29513  grpolcan  29514  grpo2inv  29515  grpoinvf  29516  grpoinvop  29517  grpodivinv  29520  grpoinvdiv  29521  grpodivf  29522  grpomuldivass  29525  grponpcan  29527  ablodivdiv4  29538  vcm  29560  rngonegcl  36432  isdrngo2  36463
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