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Theorem grpoinvval 30285
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 30284 . . 3 (𝐺 ∈ GrpOp β†’ 𝑁 = (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ)))
54fveq1d 6887 . 2 (𝐺 ∈ GrpOp β†’ (π‘β€˜π΄) = ((π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ))β€˜π΄))
6 oveq2 7413 . . . . 5 (π‘₯ = 𝐴 β†’ (𝑦𝐺π‘₯) = (𝑦𝐺𝐴))
76eqeq1d 2728 . . . 4 (π‘₯ = 𝐴 β†’ ((𝑦𝐺π‘₯) = π‘ˆ ↔ (𝑦𝐺𝐴) = π‘ˆ))
87riotabidv 7363 . . 3 (π‘₯ = 𝐴 β†’ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = π‘ˆ))
9 eqid 2726 . . 3 (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ)) = (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ))
10 riotaex 7365 . . 3 (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = π‘ˆ) ∈ V
118, 9, 10fvmpt 6992 . 2 (𝐴 ∈ 𝑋 β†’ ((π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ))β€˜π΄) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = π‘ˆ))
125, 11sylan9eq 2786 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 395   = wceq 1533   ∈ wcel 2098   ↦ cmpt 5224  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-ginv 30257
This theorem is referenced by:  grpoinvcl  30286  grpoinv  30287
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