Theorem grpoinvval 31012

Description: The inverse of a group element. (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
grpoinvval	⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈))

Distinct variable groups:   𝑦,𝐴   𝑦,𝐺   𝑦,𝑋

Allowed substitution hints:   𝑈(𝑦)   𝑁(𝑦)

Proof of Theorem grpoinvval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpinvfval.1	. . . 4 ⊢ 𝑋 = ran 𝐺
2		grpinvfval.2	. . . 4 ⊢ 𝑈 = (GId‘𝐺)
3		grpinvfval.3	. . . 4 ⊢ 𝑁 = (inv‘𝐺)
4	1, 2, 3	grpoinvfval 31011	. . 3 ⊢ (𝐺 ∈ GrpOp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))
5	4	fveq1d 6884	. 2 ⊢ (𝐺 ∈ GrpOp → (𝑁‘𝐴) = ((𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))‘𝐴))
6		oveq2 7425	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦𝐺𝑥) = (𝑦𝐺𝐴))
7	6	eqeq1d 2764	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑦𝐺𝑥) = 𝑈 ↔ (𝑦𝐺𝐴) = 𝑈))
8	7	riotabidv 7376	. . 3 ⊢ (𝑥 = 𝐴 → (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈))
9		eqid 2762	. . 3 ⊢ (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))
10		riotaex 7378	. . 3 ⊢ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈) ∈ V
11	8, 9, 10	fvmpt 6990	. 2 ⊢ (𝐴 ∈ 𝑋 → ((𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))‘𝐴) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈))
12	5, 11	sylan9eq 2817	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ↦ cmpt 5190  ran crn 5660  ‘cfv 6537  ℩crio 7373  (class class class)co 7417  GrpOpcgr 30978  GIdcgi 30979  invcgn 30980

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-ginv 30984

This theorem is used by:  grpoinvcl  31013  grpoinv  31014