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Theorem grpoinvval 27505
Description: The inverse of a group element. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpinvfval.1 𝑋 = ran 𝐺
grpinvfval.2 𝑈 = (GId‘𝐺)
grpinvfval.3 𝑁 = (inv‘𝐺)
Assertion
Ref Expression
grpoinvval ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐺   𝑦,𝑋
Allowed substitution hints:   𝑈(𝑦)   𝑁(𝑦)

Proof of Theorem grpoinvval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 grpinvfval.1 . . . 4 𝑋 = ran 𝐺
2 grpinvfval.2 . . . 4 𝑈 = (GId‘𝐺)
3 grpinvfval.3 . . . 4 𝑁 = (inv‘𝐺)
41, 2, 3grpoinvfval 27504 . . 3 (𝐺 ∈ GrpOp → 𝑁 = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
54fveq1d 6231 . 2 (𝐺 ∈ GrpOp → (𝑁𝐴) = ((𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))‘𝐴))
6 oveq2 6698 . . . . 5 (𝑥 = 𝐴 → (𝑦𝐺𝑥) = (𝑦𝐺𝐴))
76eqeq1d 2653 . . . 4 (𝑥 = 𝐴 → ((𝑦𝐺𝑥) = 𝑈 ↔ (𝑦𝐺𝐴) = 𝑈))
87riotabidv 6653 . . 3 (𝑥 = 𝐴 → (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈) = (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
9 eqid 2651 . . 3 (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))
10 riotaex 6655 . . 3 (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈) ∈ V
118, 9, 10fvmpt 6321 . 2 (𝐴𝑋 → ((𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))‘𝐴) = (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
125, 11sylan9eq 2705 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  cmpt 4762  ran crn 5144  cfv 5926  crio 6650  (class class class)co 6690  GrpOpcgr 27471  GIdcgi 27472  invcgn 27473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-ginv 27477
This theorem is referenced by:  grpoinvcl  27506  grpoinv  27507
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