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Theorem grpoinvfval 30452
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 7907 . . . . 5 (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
42, 3eqeltrid 2830 . . . 4 (𝐺 ∈ GrpOp → 𝑋 ∈ V)
5 mptexg 7230 . . . 4 (𝑋 ∈ V → (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) ∈ V)
64, 5syl 17 . . 3 (𝐺 ∈ GrpOp → (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) ∈ V)
7 rneq 5934 . . . . . 6 (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
87, 2eqtr4di 2784 . . . . 5 (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
9 oveq 7422 . . . . . . 7 (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
10 fveq2 6893 . . . . . . . 8 (𝑔 = 𝐺 → (GId‘𝑔) = (GId‘𝐺))
11 grpinvfval.2 . . . . . . . 8 𝑈 = (GId‘𝐺)
1210, 11eqtr4di 2784 . . . . . . 7 (𝑔 = 𝐺 → (GId‘𝑔) = 𝑈)
139, 12eqeq12d 2742 . . . . . 6 (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = (GId‘𝑔) ↔ (𝑦𝐺𝑥) = 𝑈))
148, 13riotaeqbidv 7375 . . . . 5 (𝑔 = 𝐺 → (𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔)) = (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))
158, 14mpteq12dv 5236 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ ran 𝑔 ↦ (𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔))) = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
16 df-ginv 30425 . . . 4 inv = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔 ↦ (𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔))))
1715, 16fvmptg 6999 . . 3 ((𝐺 ∈ GrpOp ∧ (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) ∈ V) → (inv‘𝐺) = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
186, 17mpdan 685 . 2 (𝐺 ∈ GrpOp → (inv‘𝐺) = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
191, 18eqtrid 2778 1 (𝐺 ∈ GrpOp → 𝑁 = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  Vcvv 3462  cmpt 5228  ran crn 5675  cfv 6546  crio 7371  (class class class)co 7416  GrpOpcgr 30419  GIdcgi 30420  invcgn 30421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7372  df-ov 7419  df-ginv 30425
This theorem is referenced by:  grpoinvval  30453  grpoinvf  30462
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