Theorem grpoinvcl 30599

Description: A group element's inverse is a group element. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpinvcl.1	⊢ 𝑋 = ran 𝐺
grpinvcl.2	⊢ 𝑁 = (inv‘𝐺)

Assertion

Ref	Expression
grpoinvcl	⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)

Proof of Theorem grpoinvcl

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpinvcl.1	. . 3 ⊢ 𝑋 = ran 𝐺
2		eqid 2736	. . 3 ⊢ (GId‘𝐺) = (GId‘𝐺)
3		grpinvcl.2	. . 3 ⊢ 𝑁 = (inv‘𝐺)
4	1, 2, 3	grpoinvval 30598	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GId‘𝐺)))
5	1, 2	grpoinveu 30594	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ∃!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GId‘𝐺))
6		riotacl 7332	. . 3 ⊢ (∃!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GId‘𝐺) → (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GId‘𝐺)) ∈ 𝑋)
7	5, 6	syl 17	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GId‘𝐺)) ∈ 𝑋)
8	4, 7	eqeltrd 2836	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃!wreu 3348  ran crn 5625  ‘cfv 6492  ℩crio 7314  (class class class)co 7358  GrpOpcgr 30564  GIdcgi 30565  invcgn 30566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-grpo 30568  df-gid 30569  df-ginv 30570

This theorem is referenced by:  grpoinvid1  30603  grpoinvid2  30604  grpolcan  30605  grpo2inv  30606  grpoinvf  30607  grpoinvop  30608  grpodivinv  30611  grpoinvdiv  30612  grpodivf  30613  grpomuldivass  30616  grponpcan  30618  ablodivdiv4  30629  vcm  30651  rngonegcl  38128  isdrngo2  38159