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Theorem grpoinvval 29507
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 29506 . . 3 (𝐺 ∈ GrpOp β†’ 𝑁 = (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ)))
54fveq1d 6845 . 2 (𝐺 ∈ GrpOp β†’ (π‘β€˜π΄) = ((π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ))β€˜π΄))
6 oveq2 7366 . . . . 5 (π‘₯ = 𝐴 β†’ (𝑦𝐺π‘₯) = (𝑦𝐺𝐴))
76eqeq1d 2735 . . . 4 (π‘₯ = 𝐴 β†’ ((𝑦𝐺π‘₯) = π‘ˆ ↔ (𝑦𝐺𝐴) = π‘ˆ))
87riotabidv 7316 . . 3 (π‘₯ = 𝐴 β†’ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = π‘ˆ))
9 eqid 2733 . . 3 (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ)) = (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ))
10 riotaex 7318 . . 3 (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = π‘ˆ) ∈ V
118, 9, 10fvmpt 6949 . 2 (𝐴 ∈ 𝑋 β†’ ((π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ))β€˜π΄) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = π‘ˆ))
125, 11sylan9eq 2793 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 397   = wceq 1542   ∈ wcel 2107   ↦ cmpt 5189  ran crn 5635  β€˜cfv 6497  β„©crio 7313  (class class class)co 7358  GrpOpcgr 29473  GIdcgi 29474  invcgn 29475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-ginv 29479
This theorem is referenced by:  grpoinvcl  29508  grpoinv  29509
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