Theorem grpoinvval 31158

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 31157	. . 3 ⊢ (𝐺 ∈ GrpOp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))
5	4	fveq1d 6913	. 2 ⊢ (𝐺 ∈ GrpOp → (𝑁‘𝐴) = ((𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))‘𝐴))
6		oveq2 7454	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦𝐺𝑥) = (𝑦𝐺𝐴))
7	6	eqeq1d 2765	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑦𝐺𝑥) = 𝑈 ↔ (𝑦𝐺𝐴) = 𝑈))
8	7	riotabidv 7405	. . 3 ⊢ (𝑥 = 𝐴 → (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈))
9		eqid 2763	. . 3 ⊢ (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))
10		riotaex 7407	. . 3 ⊢ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈) ∈ V
11	8, 9, 10	fvmpt 7019	. 2 ⊢ (𝐴 ∈ 𝑋 → ((𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))‘𝐴) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈))
12	5, 11	sylan9eq 2818	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ↦ cmpt 5190  ran crn 5657  ‘cfv 6561  ℩crio 7402  (class class class)co 7446  GrpOpcgr 31124  GIdcgi 31125  invcgn 31126

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pr 5396  ax-un 7778

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-ginv 31130

This theorem is used by:  grpoinvcl  31159  grpoinv  31160