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Theorem grpoinvfval 29250
Description: The inverse function of a group. (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
grpoinvfval (𝐺 ∈ GrpOp β†’ 𝑁 = (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ)))
Distinct variable groups:   π‘₯,𝑦,𝐺   π‘₯,𝑋,𝑦   π‘₯,π‘ˆ
Allowed substitution hints:   π‘ˆ(𝑦)   𝑁(π‘₯,𝑦)

Proof of Theorem grpoinvfval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 grpinvfval.3 . 2 𝑁 = (invβ€˜πΊ)
2 grpinvfval.1 . . . . 5 𝑋 = ran 𝐺
3 rnexg 7832 . . . . 5 (𝐺 ∈ GrpOp β†’ ran 𝐺 ∈ V)
42, 3eqeltrid 2843 . . . 4 (𝐺 ∈ GrpOp β†’ 𝑋 ∈ V)
5 mptexg 7166 . . . 4 (𝑋 ∈ V β†’ (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ)) ∈ V)
64, 5syl 17 . . 3 (𝐺 ∈ GrpOp β†’ (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ)) ∈ V)
7 rneq 5888 . . . . . 6 (𝑔 = 𝐺 β†’ ran 𝑔 = ran 𝐺)
87, 2eqtr4di 2796 . . . . 5 (𝑔 = 𝐺 β†’ ran 𝑔 = 𝑋)
9 oveq 7356 . . . . . . 7 (𝑔 = 𝐺 β†’ (𝑦𝑔π‘₯) = (𝑦𝐺π‘₯))
10 fveq2 6838 . . . . . . . 8 (𝑔 = 𝐺 β†’ (GIdβ€˜π‘”) = (GIdβ€˜πΊ))
11 grpinvfval.2 . . . . . . . 8 π‘ˆ = (GIdβ€˜πΊ)
1210, 11eqtr4di 2796 . . . . . . 7 (𝑔 = 𝐺 β†’ (GIdβ€˜π‘”) = π‘ˆ)
139, 12eqeq12d 2754 . . . . . 6 (𝑔 = 𝐺 β†’ ((𝑦𝑔π‘₯) = (GIdβ€˜π‘”) ↔ (𝑦𝐺π‘₯) = π‘ˆ))
148, 13riotaeqbidv 7309 . . . . 5 (𝑔 = 𝐺 β†’ (℩𝑦 ∈ ran 𝑔(𝑦𝑔π‘₯) = (GIdβ€˜π‘”)) = (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ))
158, 14mpteq12dv 5195 . . . 4 (𝑔 = 𝐺 β†’ (π‘₯ ∈ ran 𝑔 ↦ (℩𝑦 ∈ ran 𝑔(𝑦𝑔π‘₯) = (GIdβ€˜π‘”))) = (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ)))
16 df-ginv 29223 . . . 4 inv = (𝑔 ∈ GrpOp ↦ (π‘₯ ∈ ran 𝑔 ↦ (℩𝑦 ∈ ran 𝑔(𝑦𝑔π‘₯) = (GIdβ€˜π‘”))))
1715, 16fvmptg 6942 . . 3 ((𝐺 ∈ GrpOp ∧ (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ)) ∈ V) β†’ (invβ€˜πΊ) = (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ)))
186, 17mpdan 686 . 2 (𝐺 ∈ GrpOp β†’ (invβ€˜πΊ) = (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ)))
191, 18eqtrid 2790 1 (𝐺 ∈ GrpOp β†’ 𝑁 = (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3444   ↦ cmpt 5187  ran crn 5632  β€˜cfv 6492  β„©crio 7305  (class class class)co 7350  GrpOpcgr 29217  GIdcgi 29218  invcgn 29219
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 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pr 5383  ax-un 7663
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 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7306  df-ov 7353  df-ginv 29223
This theorem is referenced by:  grpoinvval  29251  grpoinvf  29260
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