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Theorem grpoinvval 27347
 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 27346 . . 3 (𝐺 ∈ GrpOp → 𝑁 = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
54fveq1d 6180 . 2 (𝐺 ∈ GrpOp → (𝑁𝐴) = ((𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))‘𝐴))
6 oveq2 6643 . . . . 5 (𝑥 = 𝐴 → (𝑦𝐺𝑥) = (𝑦𝐺𝐴))
76eqeq1d 2622 . . . 4 (𝑥 = 𝐴 → ((𝑦𝐺𝑥) = 𝑈 ↔ (𝑦𝐺𝐴) = 𝑈))
87riotabidv 6598 . . 3 (𝑥 = 𝐴 → (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈) = (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
9 eqid 2620 . . 3 (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))
10 riotaex 6600 . . 3 (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈) ∈ V
118, 9, 10fvmpt 6269 . 2 (𝐴𝑋 → ((𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))‘𝐴) = (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
125, 11sylan9eq 2674 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1481   ∈ wcel 1988   ↦ cmpt 4720  ran crn 5105  ‘cfv 5876  ℩crio 6595  (class class class)co 6635  GrpOpcgr 27313  GIdcgi 27314  invcgn 27315 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-ginv 27319 This theorem is referenced by:  grpoinvcl  27348  grpoinv  27349
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