Theorem exidresid 38124

Description: The restriction of a binary operation with identity to a subset containing the identity has the same 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
exidresid	⊢ (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) ∧ 𝐻 ∈ Magma) → (GId‘𝐻) = 𝑈)

Proof of Theorem exidresid

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

Step	Hyp	Ref	Expression
1		exidres.3	. . . . . 6 ⊢ 𝐻 = (𝐺 ↾ (𝑌 × 𝑌))
2		resexg 5994	. . . . . 6 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝐺 ↾ (𝑌 × 𝑌)) ∈ V)
3	1, 2	eqeltrid 2841	. . . . 5 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐻 ∈ V)
4		eqid 2737	. . . . . 6 ⊢ ran 𝐻 = ran 𝐻
5	4	gidval 30599	. . . . 5 ⊢ (𝐻 ∈ V → (GId‘𝐻) = (℩𝑢 ∈ ran 𝐻∀𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)))
6	3, 5	syl 17	. . . 4 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (GId‘𝐻) = (℩𝑢 ∈ ran 𝐻∀𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)))
7	6	3ad2ant1 1134	. . 3 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → (GId‘𝐻) = (℩𝑢 ∈ ran 𝐻∀𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)))
8	7	adantr 480	. 2 ⊢ (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) ∧ 𝐻 ∈ Magma) → (GId‘𝐻) = (℩𝑢 ∈ ran 𝐻∀𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)))
9		exidres.1	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
10		exidres.2	. . . . . . 7 ⊢ 𝑈 = (GId‘𝐺)
11	9, 10, 1	exidreslem 38122	. . . . . 6 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑈 ∈ dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
12	11	simprd 495	. . . . 5 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
13	12	adantr 480	. . . 4 ⊢ (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) ∧ 𝐻 ∈ Magma) → ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
14	9, 10, 1	exidres 38123	. . . . 5 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → 𝐻 ∈ ExId )
15		elin 3919	. . . . . . 7 ⊢ (𝐻 ∈ (Magma ∩ ExId ) ↔ (𝐻 ∈ Magma ∧ 𝐻 ∈ ExId ))
16		rngopidOLD 38098	. . . . . . 7 ⊢ (𝐻 ∈ (Magma ∩ ExId ) → ran 𝐻 = dom dom 𝐻)
17	15, 16	sylbir 235	. . . . . 6 ⊢ ((𝐻 ∈ Magma ∧ 𝐻 ∈ ExId ) → ran 𝐻 = dom dom 𝐻)
18	17	ancoms 458	. . . . 5 ⊢ ((𝐻 ∈ ExId ∧ 𝐻 ∈ Magma) → ran 𝐻 = dom dom 𝐻)
19	14, 18	sylan 581	. . . 4 ⊢ (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) ∧ 𝐻 ∈ Magma) → ran 𝐻 = dom dom 𝐻)
20	13, 19	raleqtrrdv 3302	. . 3 ⊢ (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) ∧ 𝐻 ∈ Magma) → ∀𝑥 ∈ ran 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
21	11	simpld 494	. . . . . 6 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → 𝑈 ∈ dom dom 𝐻)
22	21	adantr 480	. . . . 5 ⊢ (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) ∧ 𝐻 ∈ Magma) → 𝑈 ∈ dom dom 𝐻)
23	22, 19	eleqtrrd 2840	. . . 4 ⊢ (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) ∧ 𝐻 ∈ Magma) → 𝑈 ∈ ran 𝐻)
24	4	exidu1 38101	. . . . . . 7 ⊢ (𝐻 ∈ (Magma ∩ ExId ) → ∃!𝑢 ∈ ran 𝐻∀𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
25	15, 24	sylbir 235	. . . . . 6 ⊢ ((𝐻 ∈ Magma ∧ 𝐻 ∈ ExId ) → ∃!𝑢 ∈ ran 𝐻∀𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
26	25	ancoms 458	. . . . 5 ⊢ ((𝐻 ∈ ExId ∧ 𝐻 ∈ Magma) → ∃!𝑢 ∈ ran 𝐻∀𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
27	14, 26	sylan 581	. . . 4 ⊢ (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) ∧ 𝐻 ∈ Magma) → ∃!𝑢 ∈ ran 𝐻∀𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
28		oveq1 7375	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (𝑢𝐻𝑥) = (𝑈𝐻𝑥))
29	28	eqeq1d 2739	. . . . . 6 ⊢ (𝑢 = 𝑈 → ((𝑢𝐻𝑥) = 𝑥 ↔ (𝑈𝐻𝑥) = 𝑥))
30	29	ovanraleqv 7392	. . . . 5 ⊢ (𝑢 = 𝑈 → (∀𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ∀𝑥 ∈ ran 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
31	30	riota2 7350	. . . 4 ⊢ ((𝑈 ∈ ran 𝐻 ∧ ∃!𝑢 ∈ ran 𝐻∀𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)) → (∀𝑥 ∈ ran 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥) ↔ (℩𝑢 ∈ ran 𝐻∀𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)) = 𝑈))
32	23, 27, 31	syl2anc 585	. . 3 ⊢ (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) ∧ 𝐻 ∈ Magma) → (∀𝑥 ∈ ran 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥) ↔ (℩𝑢 ∈ ran 𝐻∀𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)) = 𝑈))
33	20, 32	mpbid 232	. 2 ⊢ (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) ∧ 𝐻 ∈ Magma) → (℩𝑢 ∈ ran 𝐻∀𝑥 ∈ ran 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)) = 𝑈)
34	8, 33	eqtrd 2772	1 ⊢ (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) ∧ 𝐻 ∈ Magma) → (GId‘𝐻) = 𝑈)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃!wreu 3350  Vcvv 3442   ∩ cin 3902   ⊆ wss 3903   × cxp 5630  dom cdm 5632  ran crn 5633   ↾ cres 5634  ‘cfv 6500  ℩crio 7324  (class class class)co 7368  GIdcgi 30577   ExId cexid 38089  Magmacmagm 38093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-riota 7325  df-ov 7371  df-gid 30581  df-exid 38090  df-mgmOLD 38094

This theorem is referenced by:  isdrngo2  38203