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Theorem grpoinvfval 27222
 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 7045 . . . . 5 (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
42, 3syl5eqel 2702 . . . 4 (𝐺 ∈ GrpOp → 𝑋 ∈ V)
5 mptexg 6438 . . . 4 (𝑋 ∈ V → (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) ∈ V)
64, 5syl 17 . . 3 (𝐺 ∈ GrpOp → (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) ∈ V)
7 rneq 5311 . . . . . 6 (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
87, 2syl6eqr 2673 . . . . 5 (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
9 oveq 6610 . . . . . . 7 (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
10 fveq2 6148 . . . . . . . 8 (𝑔 = 𝐺 → (GId‘𝑔) = (GId‘𝐺))
11 grpinvfval.2 . . . . . . . 8 𝑈 = (GId‘𝐺)
1210, 11syl6eqr 2673 . . . . . . 7 (𝑔 = 𝐺 → (GId‘𝑔) = 𝑈)
139, 12eqeq12d 2636 . . . . . 6 (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = (GId‘𝑔) ↔ (𝑦𝐺𝑥) = 𝑈))
148, 13riotaeqbidv 6568 . . . . 5 (𝑔 = 𝐺 → (𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔)) = (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))
158, 14mpteq12dv 4693 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ ran 𝑔 ↦ (𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔))) = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
16 df-ginv 27195 . . . 4 inv = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔 ↦ (𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔))))
1715, 16fvmptg 6237 . . 3 ((𝐺 ∈ GrpOp ∧ (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) ∈ V) → (inv‘𝐺) = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
186, 17mpdan 701 . 2 (𝐺 ∈ GrpOp → (inv‘𝐺) = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
191, 18syl5eq 2667 1 (𝐺 ∈ GrpOp → 𝑁 = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480   ∈ wcel 1987  Vcvv 3186   ↦ cmpt 4673  ran crn 5075  ‘cfv 5847  ℩crio 6564  (class class class)co 6604  GrpOpcgr 27189  GIdcgi 27190  invcgn 27191 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-ginv 27195 This theorem is referenced by:  grpoinvval  27223  grpoinvf  27232
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