Theorem grpoinvcl 30504

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 2731	. . 3 ⊢ (GId‘𝐺) = (GId‘𝐺)
3		grpinvcl.2	. . 3 ⊢ 𝑁 = (inv‘𝐺)
4	1, 2, 3	grpoinvval 30503	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GId‘𝐺)))
5	1, 2	grpoinveu 30499	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ∃!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GId‘𝐺))
6		riotacl 7320	. . 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 2831	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃!wreu 3344  ran crn 5615  ‘cfv 6481  ℩crio 7302  (class class class)co 7346  GrpOpcgr 30469  GIdcgi 30470  invcgn 30471

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-grpo 30473  df-gid 30474  df-ginv 30475

This theorem is referenced by:  grpoinvid1  30508  grpoinvid2  30509  grpolcan  30510  grpo2inv  30511  grpoinvf  30512  grpoinvop  30513  grpodivinv  30516  grpoinvdiv  30517  grpodivf  30518  grpomuldivass  30521  grponpcan  30523  ablodivdiv4  30534  vcm  30556  rngonegcl  37977  isdrngo2  38008