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Theorem grpoinvcl 30286
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 2726 . . 3 (GIdβ€˜πΊ) = (GIdβ€˜πΊ)
3 grpinvcl.2 . . 3 𝑁 = (invβ€˜πΊ)
41, 2, 3grpoinvval 30285 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GIdβ€˜πΊ)))
51, 2grpoinveu 30281 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ βˆƒ!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GIdβ€˜πΊ))
6 riotacl 7379 . . 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 2827 1 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆƒ!wreu 3368  ran crn 5670  β€˜cfv 6537  β„©crio 7360  (class class class)co 7405  GrpOpcgr 30251  GIdcgi 30252  invcgn 30253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-grpo 30255  df-gid 30256  df-ginv 30257
This theorem is referenced by:  grpoinvid1  30290  grpoinvid2  30291  grpolcan  30292  grpo2inv  30293  grpoinvf  30294  grpoinvop  30295  grpodivinv  30298  grpoinvdiv  30299  grpodivf  30300  grpomuldivass  30303  grponpcan  30305  ablodivdiv4  30316  vcm  30338  rngonegcl  37308  isdrngo2  37339
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