Theorem grpoinvfval 30609

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 7854	. . . . 5 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
4	2, 3	eqeltrid 2841	. . . 4 ⊢ (𝐺 ∈ GrpOp → 𝑋 ∈ V)
5		mptexg 7177	. . . 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 5893	. . . . . 6 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
8	7, 2	eqtr4di 2790	. . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
9		oveq 7374	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
10		fveq2 6842	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (GId‘𝑔) = (GId‘𝐺))
11		grpinvfval.2	. . . . . . . 8 ⊢ 𝑈 = (GId‘𝐺)
12	10, 11	eqtr4di 2790	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (GId‘𝑔) = 𝑈)
13	9, 12	eqeq12d 2753	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = (GId‘𝑔) ↔ (𝑦𝐺𝑥) = 𝑈))
14	8, 13	riotaeqbidv 7328	. . . . 5 ⊢ (𝑔 = 𝐺 → (℩𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔)) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))
15	8, 14	mpteq12dv 5187	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ ran 𝑔 ↦ (℩𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔))) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))
16		df-ginv 30582	. . . 4 ⊢ inv = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔 ↦ (℩𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔))))
17	15, 16	fvmptg 6947	. . 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 2784	1 ⊢ (𝐺 ∈ GrpOp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ↦ cmpt 5181  ran crn 5633  ‘cfv 6500  ℩crio 7324  (class class class)co 7368  GrpOpcgr 30576  GIdcgi 30577  invcgn 30578

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 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-ginv 30582

This theorem is referenced by:  grpoinvval  30610  grpoinvf  30619