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Theorem grpoinvfval 29262
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 7831 . . . . 5 (𝐺 ∈ GrpOp β†’ ran 𝐺 ∈ V)
42, 3eqeltrid 2842 . . . 4 (𝐺 ∈ GrpOp β†’ 𝑋 ∈ V)
5 mptexg 7165 . . . 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 5887 . . . . . 6 (𝑔 = 𝐺 β†’ ran 𝑔 = ran 𝐺)
87, 2eqtr4di 2795 . . . . 5 (𝑔 = 𝐺 β†’ ran 𝑔 = 𝑋)
9 oveq 7355 . . . . . . 7 (𝑔 = 𝐺 β†’ (𝑦𝑔π‘₯) = (𝑦𝐺π‘₯))
10 fveq2 6837 . . . . . . . 8 (𝑔 = 𝐺 β†’ (GIdβ€˜π‘”) = (GIdβ€˜πΊ))
11 grpinvfval.2 . . . . . . . 8 π‘ˆ = (GIdβ€˜πΊ)
1210, 11eqtr4di 2795 . . . . . . 7 (𝑔 = 𝐺 β†’ (GIdβ€˜π‘”) = π‘ˆ)
139, 12eqeq12d 2753 . . . . . 6 (𝑔 = 𝐺 β†’ ((𝑦𝑔π‘₯) = (GIdβ€˜π‘”) ↔ (𝑦𝐺π‘₯) = π‘ˆ))
148, 13riotaeqbidv 7308 . . . . 5 (𝑔 = 𝐺 β†’ (℩𝑦 ∈ ran 𝑔(𝑦𝑔π‘₯) = (GIdβ€˜π‘”)) = (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ))
158, 14mpteq12dv 5194 . . . 4 (𝑔 = 𝐺 β†’ (π‘₯ ∈ ran 𝑔 ↦ (℩𝑦 ∈ ran 𝑔(𝑦𝑔π‘₯) = (GIdβ€˜π‘”))) = (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ)))
16 df-ginv 29235 . . . 4 inv = (𝑔 ∈ GrpOp ↦ (π‘₯ ∈ ran 𝑔 ↦ (℩𝑦 ∈ ran 𝑔(𝑦𝑔π‘₯) = (GIdβ€˜π‘”))))
1715, 16fvmptg 6941 . . 3 ((𝐺 ∈ GrpOp ∧ (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ)) ∈ V) β†’ (invβ€˜πΊ) = (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ)))
186, 17mpdan 685 . 2 (𝐺 ∈ GrpOp β†’ (invβ€˜πΊ) = (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ)))
191, 18eqtrid 2789 1 (𝐺 ∈ GrpOp β†’ 𝑁 = (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3443   ↦ cmpt 5186  ran crn 5631  β€˜cfv 6491  β„©crio 7304  (class class class)co 7349  GrpOpcgr 29229  GIdcgi 29230  invcgn 29231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7662
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5528  df-xp 5636  df-rel 5637  df-cnv 5638  df-co 5639  df-dm 5640  df-rn 5641  df-res 5642  df-ima 5643  df-iota 6443  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7305  df-ov 7352  df-ginv 29235
This theorem is referenced by:  grpoinvval  29263  grpoinvf  29272
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