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Theorem grpoinvfval 27606
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 7215 . . . . 5 (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
42, 3syl5eqel 2807 . . . 4 (𝐺 ∈ GrpOp → 𝑋 ∈ V)
5 mptexg 6600 . . . 4 (𝑋 ∈ V → (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) ∈ V)
64, 5syl 17 . . 3 (𝐺 ∈ GrpOp → (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) ∈ V)
7 rneq 5458 . . . . . 6 (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
87, 2syl6eqr 2776 . . . . 5 (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
9 oveq 6771 . . . . . . 7 (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
10 fveq2 6304 . . . . . . . 8 (𝑔 = 𝐺 → (GId‘𝑔) = (GId‘𝐺))
11 grpinvfval.2 . . . . . . . 8 𝑈 = (GId‘𝐺)
1210, 11syl6eqr 2776 . . . . . . 7 (𝑔 = 𝐺 → (GId‘𝑔) = 𝑈)
139, 12eqeq12d 2739 . . . . . 6 (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = (GId‘𝑔) ↔ (𝑦𝐺𝑥) = 𝑈))
148, 13riotaeqbidv 6729 . . . . 5 (𝑔 = 𝐺 → (𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔)) = (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))
158, 14mpteq12dv 4841 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ ran 𝑔 ↦ (𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔))) = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
16 df-ginv 27579 . . . 4 inv = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔 ↦ (𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔))))
1715, 16fvmptg 6394 . . 3 ((𝐺 ∈ GrpOp ∧ (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) ∈ V) → (inv‘𝐺) = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
186, 17mpdan 705 . 2 (𝐺 ∈ GrpOp → (inv‘𝐺) = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
191, 18syl5eq 2770 1 (𝐺 ∈ GrpOp → 𝑁 = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1596  wcel 2103  Vcvv 3304  cmpt 4837  ran crn 5219  cfv 6001  crio 6725  (class class class)co 6765  GrpOpcgr 27573  GIdcgi 27574  invcgn 27575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-ginv 27579
This theorem is referenced by:  grpoinvval  27607  grpoinvf  27616
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