Theorem ismndo 38625

Description: Obsolete theorem, use ismnddef 18841 instead. The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
ismndo.1	⊢ 𝑋 = dom dom 𝐺

Assertion

Ref	Expression
ismndo	⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ MndOp ↔ (𝐺 ∈ SemiGrp ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))))

Distinct variable groups:   𝑥,𝐺,𝑦   𝑥,𝑋,𝑦

Allowed substitution hints:   𝐴(𝑥, 𝑦)

Proof of Theorem ismndo

Step	Hyp	Ref	Expression
1		df-mndo 38620	. . 3 ⊢ MndOp = (SemiGrp ∩ ExId )
2	1	eleq2i 2852	. 2 ⊢ (𝐺 ∈ MndOp ↔ 𝐺 ∈ (SemiGrp ∩ ExId ))
3		elin 3915	. . 3 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) ↔ (𝐺 ∈ SemiGrp ∧ 𝐺 ∈ ExId ))
4		ismndo.1	. . . . 5 ⊢ 𝑋 = dom dom 𝐺
5	4	isexid 38600	. . . 4 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ ExId ↔ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
6	5	anbi2d 642	. . 3 ⊢ (𝐺 ∈ 𝐴 → ((𝐺 ∈ SemiGrp ∧ 𝐺 ∈ ExId ) ↔ (𝐺 ∈ SemiGrp ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))))
7	3, 6	bitrid 286	. 2 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ (SemiGrp ∩ ExId ) ↔ (𝐺 ∈ SemiGrp ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))))
8	2, 7	bitrid 286	1 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ MndOp ↔ (𝐺 ∈ SemiGrp ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   ∩ cin 3898  dom cdm 5655  (class class class)co 7414   ExId cexid 38597  SemiGrpcsem 38613  MndOpcmndo 38619

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

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-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-dm 5665  df-iota 6489  df-fv 6541  df-ov 7417  df-exid 38598  df-mndo 38620

This theorem is used by:  ismndo1  38626