Theorem grpoinvfval 30593

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 7853	. . . . 5 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
4	2, 3	eqeltrid 2840	. . . 4 ⊢ (𝐺 ∈ GrpOp → 𝑋 ∈ V)
5		mptexg 7176	. . . 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 5891	. . . . . 6 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
8	7, 2	eqtr4di 2789	. . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
9		oveq 7373	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
10		fveq2 6840	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (GId‘𝑔) = (GId‘𝐺))
11		grpinvfval.2	. . . . . . . 8 ⊢ 𝑈 = (GId‘𝐺)
12	10, 11	eqtr4di 2789	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (GId‘𝑔) = 𝑈)
13	9, 12	eqeq12d 2752	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = (GId‘𝑔) ↔ (𝑦𝐺𝑥) = 𝑈))
14	8, 13	riotaeqbidv 7327	. . . . 5 ⊢ (𝑔 = 𝐺 → (℩𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔)) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))
15	8, 14	mpteq12dv 5172	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ ran 𝑔 ↦ (℩𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔))) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))
16		df-ginv 30566	. . . 4 ⊢ inv = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔 ↦ (℩𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔))))
17	15, 16	fvmptg 6945	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)) ∈ V) → (inv‘𝐺) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))
18	6, 17	mpdan 688	. 2 ⊢ (𝐺 ∈ GrpOp → (inv‘𝐺) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))
19	1, 18	eqtrid 2783	1 ⊢ (𝐺 ∈ GrpOp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ↦ cmpt 5166  ran crn 5632  ‘cfv 6498  ℩crio 7323  (class class class)co 7367  GrpOpcgr 30560  GIdcgi 30561  invcgn 30562

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-ginv 30566

This theorem is referenced by:  grpoinvval  30594  grpoinvf  30603