Theorem exidres 35963

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 35962	. . 3 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑈 ∈ dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
5		oveq1 7262	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢𝐻𝑥) = (𝑈𝐻𝑥))
6	5	eqeq1d 2740	. . . . 5 ⊢ (𝑢 = 𝑈 → ((𝑢𝐻𝑥) = 𝑥 ↔ (𝑈𝐻𝑥) = 𝑥))
7	6	ovanraleqv 7279	. . . 4 ⊢ (𝑢 = 𝑈 → (∀𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
8	7	rspcev 3552	. . 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 5926	. . . . 5 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝐺 ↾ (𝑌 × 𝑌)) ∈ V)
11	3, 10	eqeltrid 2843	. . . 4 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐻 ∈ V)
12		eqid 2738	. . . . 5 ⊢ dom dom 𝐻 = dom dom 𝐻
13	12	isexid 35932	. . . 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 1131	. 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 256	1 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → 𝐻 ∈ ExId )

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883   × cxp 5578  dom cdm 5580  ran crn 5581   ↾ cres 5582  ‘cfv 6418  (class class class)co 7255  GIdcgi 28753   ExId cexid 35929  Magmacmagm 35933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fo 6424  df-fv 6426  df-riota 7212  df-ov 7258  df-gid 28757  df-exid 35930  df-mgmOLD 35934

This theorem is referenced by:  exidresid  35964