Theorem grpoinvval 30451

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 30450	. . 3 ⊢ (𝐺 ∈ GrpOp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))
5	4	fveq1d 6893	. 2 ⊢ (𝐺 ∈ GrpOp → (𝑁‘𝐴) = ((𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))‘𝐴))
6		oveq2 7422	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦𝐺𝑥) = (𝑦𝐺𝐴))
7	6	eqeq1d 2728	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑦𝐺𝑥) = 𝑈 ↔ (𝑦𝐺𝐴) = 𝑈))
8	7	riotabidv 7372	. . 3 ⊢ (𝑥 = 𝐴 → (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈))
9		eqid 2726	. . 3 ⊢ (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))
10		riotaex 7374	. . 3 ⊢ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈) ∈ V
11	8, 9, 10	fvmpt 6999	. 2 ⊢ (𝐴 ∈ 𝑋 → ((𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))‘𝐴) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈))
12	5, 11	sylan9eq 2786	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534   ∈ wcel 2099   ↦ cmpt 5227  ran crn 5674  ‘cfv 6544  ℩crio 7369  (class class class)co 7414  GrpOpcgr 30417  GIdcgi 30418  invcgn 30419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pr 5424  ax-un 7736

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7370  df-ov 7417  df-ginv 30423

This theorem is referenced by:  grpoinvcl  30452  grpoinv  30453