Theorem exidres 36746

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 36745	. . 3 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑈 ∈ dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
5		oveq1 7416	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢𝐻𝑥) = (𝑈𝐻𝑥))
6	5	eqeq1d 2735	. . . . 5 ⊢ (𝑢 = 𝑈 → ((𝑢𝐻𝑥) = 𝑥 ↔ (𝑈𝐻𝑥) = 𝑥))
7	6	ovanraleqv 7433	. . . 4 ⊢ (𝑢 = 𝑈 → (∀𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
8	7	rspcev 3613	. . 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 6028	. . . . 5 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝐺 ↾ (𝑌 × 𝑌)) ∈ V)
11	3, 10	eqeltrid 2838	. . . 4 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐻 ∈ V)
12		eqid 2733	. . . . 5 ⊢ dom dom 𝐻 = dom dom 𝐻
13	12	isexid 36715	. . . 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 1134	. 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 257	1 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → 𝐻 ∈ ExId )

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071  Vcvv 3475   ∩ cin 3948   ⊆ wss 3949   × cxp 5675  dom cdm 5677  ran crn 5678   ↾ cres 5679  ‘cfv 6544  (class class class)co 7409  GIdcgi 29743   ExId cexid 36712  Magmacmagm 36716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-riota 7365  df-ov 7412  df-gid 29747  df-exid 36713  df-mgmOLD 36717

This theorem is referenced by:  exidresid  36747