Theorem grpoinvfval 28884

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 = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (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.

Step	Hyp	Ref	Expression
1		grpinvfval.3	. 2 ⊢ 𝑁 = (inv‘𝐺)
2		grpinvfval.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
3		rnexg 7751	. . . . 5 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
4	2, 3	eqeltrid 2843	. . . 4 ⊢ (𝐺 ∈ GrpOp → 𝑋 ∈ V)
5		mptexg 7097	. . . 4 ⊢ (𝑋 ∈ V → (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)) ∈ V)
6	4, 5	syl 17	. . 3 ⊢ (𝐺 ∈ GrpOp → (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)) ∈ V)
7		rneq 5845	. . . . . 6 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
8	7, 2	eqtr4di 2796	. . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
9		oveq 7281	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
10		fveq2 6774	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (GId‘𝑔) = (GId‘𝐺))
11		grpinvfval.2	. . . . . . . 8 ⊢ 𝑈 = (GId‘𝐺)
12	10, 11	eqtr4di 2796	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (GId‘𝑔) = 𝑈)
13	9, 12	eqeq12d 2754	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = (GId‘𝑔) ↔ (𝑦𝐺𝑥) = 𝑈))
14	8, 13	riotaeqbidv 7235	. . . . 5 ⊢ (𝑔 = 𝐺 → (℩𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔)) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))
15	8, 14	mpteq12dv 5165	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ ran 𝑔 ↦ (℩𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔))) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))
16		df-ginv 28857	. . . 4 ⊢ inv = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔 ↦ (℩𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔))))
17	15, 16	fvmptg 6873	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)) ∈ V) → (inv‘𝐺) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))
18	6, 17	mpdan 684	. 2 ⊢ (𝐺 ∈ GrpOp → (inv‘𝐺) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))
19	1, 18	eqtrid 2790	1 ⊢ (𝐺 ∈ GrpOp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ↦ cmpt 5157  ran crn 5590  ‘cfv 6433  ℩crio 7231  (class class class)co 7275  GrpOpcgr 28851  GIdcgi 28852  invcgn 28853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-ginv 28857

This theorem is referenced by:  grpoinvval  28885  grpoinvf  28894