Theorem grpoidcl 30495

Description: The identity element of a group belongs to the group. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpoidval.1	⊢ 𝑋 = ran 𝐺
grpoidval.2	⊢ 𝑈 = (GId‘𝐺)

Assertion

Ref	Expression
grpoidcl	⊢ (𝐺 ∈ GrpOp → 𝑈 ∈ 𝑋)

Proof of Theorem grpoidcl

Dummy variables 𝑥 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpoidval.1	. . 3 ⊢ 𝑋 = ran 𝐺
2		grpoidval.2	. . 3 ⊢ 𝑈 = (GId‘𝐺)
3	1, 2	grpoidval 30494	. 2 ⊢ (𝐺 ∈ GrpOp → 𝑈 = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥))
4	1	grpoideu 30490	. . 3 ⊢ (𝐺 ∈ GrpOp → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
5		riotacl 7379	. . 3 ⊢ (∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 → (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥) ∈ 𝑋)
6	4, 5	syl 17	. 2 ⊢ (𝐺 ∈ GrpOp → (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥) ∈ 𝑋)
7	3, 6	eqeltrd 2834	1 ⊢ (𝐺 ∈ GrpOp → 𝑈 ∈ 𝑋)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃!wreu 3357  ran crn 5655  ‘cfv 6531  ℩crio 7361  (class class class)co 7405  GrpOpcgr 30470  GIdcgi 30471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fo 6537  df-fv 6539  df-riota 7362  df-ov 7408  df-grpo 30474  df-gid 30475

This theorem is referenced by:  grpoid  30501  vczcl  30553  nvzcl  30615  ghomidOLD  37913  grpokerinj  37917  rngo0cl  37943  rngolz  37946  rngorz  37947  gidsn  37976  keridl  38056