Theorem grpoidcl 30771

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 30770	. 2 ⊢ (𝐺 ∈ GrpOp → 𝑈 = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥))
4	1	grpoideu 30766	. . 3 ⊢ (𝐺 ∈ GrpOp → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
5		riotacl 7374	. . 3 ⊢ (∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 → (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥) ∈ 𝑋)
6	4, 5	syl 18	. 2 ⊢ (𝐺 ∈ GrpOp → (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥) ∈ 𝑋)
7	3, 6	eqeltrd 2865	1 ⊢ (𝐺 ∈ GrpOp → 𝑈 ∈ 𝑋)

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ∃!wreu 3368  ran crn 5652  ‘cfv 6525  ℩crio 7356  (class class class)co 7400  GrpOpcgr 30746  GIdcgi 30747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fo 6531  df-fv 6533  df-riota 7357  df-ov 7403  df-grpo 30750  df-gid 30751

This theorem is referenced by:  grpoid  30777  vczcl  30829  nvzcl  30891  ghomidOLD  38395  grpokerinj  38399  rngo0cl  38425  rngolz  38428  rngorz  38429  gidsn  38458  keridl  38538