Theorem exidres 35773

Description: The restriction of a binary operation with identity to a subset containing the identity has an identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)

Hypotheses

Ref	Expression
exidres.1	⊢ 𝑋 = ran 𝐺
exidres.2	⊢ 𝑈 = (GId‘𝐺)
exidres.3	⊢ 𝐻 = (𝐺 ↾ (𝑌 × 𝑌))

Assertion

Ref	Expression
exidres	⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → 𝐻 ∈ ExId )

Proof of Theorem exidres

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

Step	Hyp	Ref	Expression
1		exidres.1	. . . 4 ⊢ 𝑋 = ran 𝐺
2		exidres.2	. . . 4 ⊢ 𝑈 = (GId‘𝐺)
3		exidres.3	. . . 4 ⊢ 𝐻 = (𝐺 ↾ (𝑌 × 𝑌))
4	1, 2, 3	exidreslem 35772	. . 3 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑈 ∈ dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
5		oveq1 7220	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢𝐻𝑥) = (𝑈𝐻𝑥))
6	5	eqeq1d 2739	. . . . 5 ⊢ (𝑢 = 𝑈 → ((𝑢𝐻𝑥) = 𝑥 ↔ (𝑈𝐻𝑥) = 𝑥))
7	6	ovanraleqv 7237	. . . 4 ⊢ (𝑢 = 𝑈 → (∀𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
8	7	rspcev 3537	. . 3 ⊢ ((𝑈 ∈ dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)) → ∃𝑢 ∈ dom dom 𝐻∀𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
9	4, 8	syl 17	. 2 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → ∃𝑢 ∈ dom dom 𝐻∀𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
10		resexg 5897	. . . . 5 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝐺 ↾ (𝑌 × 𝑌)) ∈ V)
11	3, 10	eqeltrid 2842	. . . 4 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐻 ∈ V)
12		eqid 2737	. . . . 5 ⊢ dom dom 𝐻 = dom dom 𝐻
13	12	isexid 35742	. . . 4 ⊢ (𝐻 ∈ V → (𝐻 ∈ ExId ↔ ∃𝑢 ∈ dom dom 𝐻∀𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)))
14	11, 13	syl 17	. . 3 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝐻 ∈ ExId ↔ ∃𝑢 ∈ dom dom 𝐻∀𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)))
15	14	3ad2ant1 1135	. 2 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝐻 ∈ ExId ↔ ∃𝑢 ∈ dom dom 𝐻∀𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)))
16	9, 15	mpbird 260	1 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → 𝐻 ∈ ExId )

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∀wral 3061  ∃wrex 3062  Vcvv 3408   ∩ cin 3865   ⊆ wss 3866   × cxp 5549  dom cdm 5551  ran crn 5552   ↾ cres 5553  ‘cfv 6380  (class class class)co 7213  GIdcgi 28571   ExId cexid 35739  Magmacmagm 35743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fo 6386  df-fv 6388  df-riota 7170  df-ov 7216  df-gid 28575  df-exid 35740  df-mgmOLD 35744

This theorem is referenced by:  exidresid  35774