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Theorem grpoinvval 30377
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 π‘ˆ = (GIdβ€˜πΊ)
grpinvfval.3 𝑁 = (invβ€˜πΊ)
Assertion
Ref Expression
grpoinvval ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = π‘ˆ))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐺   𝑦,𝑋
Allowed substitution hints:   π‘ˆ(𝑦)   𝑁(𝑦)

Proof of Theorem grpoinvval
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 grpinvfval.1 . . . 4 𝑋 = ran 𝐺
2 grpinvfval.2 . . . 4 π‘ˆ = (GIdβ€˜πΊ)
3 grpinvfval.3 . . . 4 𝑁 = (invβ€˜πΊ)
41, 2, 3grpoinvfval 30376 . . 3 (𝐺 ∈ GrpOp β†’ 𝑁 = (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ)))
54fveq1d 6894 . 2 (𝐺 ∈ GrpOp β†’ (π‘β€˜π΄) = ((π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ))β€˜π΄))
6 oveq2 7424 . . . . 5 (π‘₯ = 𝐴 β†’ (𝑦𝐺π‘₯) = (𝑦𝐺𝐴))
76eqeq1d 2727 . . . 4 (π‘₯ = 𝐴 β†’ ((𝑦𝐺π‘₯) = π‘ˆ ↔ (𝑦𝐺𝐴) = π‘ˆ))
87riotabidv 7374 . . 3 (π‘₯ = 𝐴 β†’ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = π‘ˆ))
9 eqid 2725 . . 3 (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ)) = (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ))
10 riotaex 7376 . . 3 (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = π‘ˆ) ∈ V
118, 9, 10fvmpt 7000 . 2 (𝐴 ∈ 𝑋 β†’ ((π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ))β€˜π΄) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = π‘ˆ))
125, 11sylan9eq 2785 1 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = π‘ˆ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ↦ cmpt 5226  ran crn 5673  β€˜cfv 6543  β„©crio 7371  (class class class)co 7416  GrpOpcgr 30343  GIdcgi 30344  invcgn 30345
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 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-ginv 30349
This theorem is referenced by:  grpoinvcl  30378  grpoinv  30379
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