Theorem iss 6039

Description: A subclass of the identity function is the identity function restricted to its domain. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
iss	⊢ (𝐴 ⊆ I ↔ 𝐴 = ( I ↾ dom 𝐴))

Proof of Theorem iss

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

Step	Hyp	Ref	Expression
1		vex 3461	. . . . . . . . 9 ⊢ 𝑥 ∈ V
2		vex 3461	. . . . . . . . 9 ⊢ 𝑦 ∈ V
3	1, 2	opeldm 5899	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
4	3	a1i 11	. . . . . . 7 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴))
5		ssel 3932	. . . . . . 7 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ I ))
6	4, 5	jcad 522	. . . . . 6 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I )))
7		df-br 5112	. . . . . . . . 9 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
8	2	ideq 5840	. . . . . . . . 9 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
9	7, 8	bitr3i 280	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
10	1	eldm2 5893	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
11		opeq2 4841	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
12	11	eleq1d 2850	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (⟨𝑥, 𝑥⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
13	12	biimprcd 253	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
14	9, 13	biimtrid 245	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
15	5, 14	sylcom 31	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
16	15	exlimdv 1966	. . . . . . . . . 10 ⊢ (𝐴 ⊆ I → (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
17	10, 16	biimtrid 245	. . . . . . . . 9 ⊢ (𝐴 ⊆ I → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
18	12	imbi2d 343	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴) ↔ (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
19	17, 18	syl5ibcom 248	. . . . . . . 8 ⊢ (𝐴 ⊆ I → (𝑥 = 𝑦 → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
20	9, 19	biimtrid 245	. . . . . . 7 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ I → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
21	20	impcomd 417	. . . . . 6 ⊢ (𝐴 ⊆ I → ((𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) → ⟨𝑥, 𝑦⟩ ∈ 𝐴))
22	6, 21	impbid 215	. . . . 5 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I )))
23	2	opelresi 5988	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴) ↔ (𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
24	22, 23	bitr4di 292	. . . 4 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴)))
25	24	alrimivv 1961	. . 3 ⊢ (𝐴 ⊆ I → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴)))
26		reli 5815	. . . . 5 ⊢ Rel I
27		relss 5770	. . . . 5 ⊢ (𝐴 ⊆ I → (Rel I → Rel 𝐴))
28	26, 27	mpi 21	. . . 4 ⊢ (𝐴 ⊆ I → Rel 𝐴)
29		relres 6006	. . . 4 ⊢ Rel ( I ↾ dom 𝐴)
30		eqrel 5772	. . . 4 ⊢ ((Rel 𝐴 ∧ Rel ( I ↾ dom 𝐴)) → (𝐴 = ( I ↾ dom 𝐴) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴))))
31	28, 29, 30	sylancl 598	. . 3 ⊢ (𝐴 ⊆ I → (𝐴 = ( I ↾ dom 𝐴) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴))))
32	25, 31	mpbird 260	. 2 ⊢ (𝐴 ⊆ I → 𝐴 = ( I ↾ dom 𝐴))
33		resss 6002	. . 3 ⊢ ( I ↾ dom 𝐴) ⊆ I
34		sseq1 3963	. . 3 ⊢ (𝐴 = ( I ↾ dom 𝐴) → (𝐴 ⊆ I ↔ ( I ↾ dom 𝐴) ⊆ I ))
35	33, 34	mpbiri 261	. 2 ⊢ (𝐴 = ( I ↾ dom 𝐴) → 𝐴 ⊆ I )
36	32, 35	impbii 212	1 ⊢ (𝐴 ⊆ I ↔ 𝐴 = ( I ↾ dom 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2146   ⊆ wss 3906  ⟨cop 4597   class class class wbr 5111   I cid 5557  dom cdm 5663   ↾ cres 5665  Rel wrel 5668

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 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

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 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-dm 5673  df-res 5675

This theorem is used by:  funcocnv2  6850  trust  24415