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Theorem grpoinvfval 28303
 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 𝑈 = (GId‘𝐺)
grpinvfval.3 𝑁 = (inv‘𝐺)
Assertion
Ref Expression
grpoinvfval (𝐺 ∈ GrpOp → 𝑁 = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑋,𝑦   𝑥,𝑈
Allowed substitution hints:   𝑈(𝑦)   𝑁(𝑥,𝑦)

Proof of Theorem grpoinvfval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 grpinvfval.3 . 2 𝑁 = (inv‘𝐺)
2 grpinvfval.1 . . . . 5 𝑋 = ran 𝐺
3 rnexg 7600 . . . . 5 (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
42, 3eqeltrid 2918 . . . 4 (𝐺 ∈ GrpOp → 𝑋 ∈ V)
5 mptexg 6966 . . . 4 (𝑋 ∈ V → (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) ∈ V)
64, 5syl 17 . . 3 (𝐺 ∈ GrpOp → (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) ∈ V)
7 rneq 5783 . . . . . 6 (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
87, 2eqtr4di 2875 . . . . 5 (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
9 oveq 7146 . . . . . . 7 (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
10 fveq2 6652 . . . . . . . 8 (𝑔 = 𝐺 → (GId‘𝑔) = (GId‘𝐺))
11 grpinvfval.2 . . . . . . . 8 𝑈 = (GId‘𝐺)
1210, 11eqtr4di 2875 . . . . . . 7 (𝑔 = 𝐺 → (GId‘𝑔) = 𝑈)
139, 12eqeq12d 2838 . . . . . 6 (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = (GId‘𝑔) ↔ (𝑦𝐺𝑥) = 𝑈))
148, 13riotaeqbidv 7101 . . . . 5 (𝑔 = 𝐺 → (𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔)) = (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))
158, 14mpteq12dv 5127 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ ran 𝑔 ↦ (𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔))) = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
16 df-ginv 28276 . . . 4 inv = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔 ↦ (𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔))))
1715, 16fvmptg 6748 . . 3 ((𝐺 ∈ GrpOp ∧ (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) ∈ V) → (inv‘𝐺) = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
186, 17mpdan 686 . 2 (𝐺 ∈ GrpOp → (inv‘𝐺) = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
191, 18syl5eq 2869 1 (𝐺 ∈ GrpOp → 𝑁 = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2114  Vcvv 3469   ↦ cmpt 5122  ran crn 5533  ‘cfv 6334  ℩crio 7097  (class class class)co 7140  GrpOpcgr 28270  GIdcgi 28271  invcgn 28272 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-ginv 28276 This theorem is referenced by:  grpoinvval  28304  grpoinvf  28313
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