Theorem iss 4930

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		ssel 3136	. . . . . . 7 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ I ))
2		vex 2729	. . . . . . . . 9 ⊢ 𝑥 ∈ V
3		vex 2729	. . . . . . . . 9 ⊢ 𝑦 ∈ V
4	2, 3	opeldm 4807	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
5	4	a1i 9	. . . . . . 7 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴))
6	1, 5	jcad 305	. . . . . 6 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴)))
7		df-br 3983	. . . . . . . . 9 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
8	3	ideq 4756	. . . . . . . . 9 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
9	7, 8	bitr3i 185	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
10	2	eldm2 4802	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
11		opeq2 3759	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
12	11	eleq1d 2235	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (⟨𝑥, 𝑥⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
13	12	biimprcd 159	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
14	9, 13	syl5bi 151	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
15	1, 14	sylcom 28	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
16	15	exlimdv 1807	. . . . . . . . . 10 ⊢ (𝐴 ⊆ I → (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
17	10, 16	syl5bi 151	. . . . . . . . 9 ⊢ (𝐴 ⊆ I → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
18	12	imbi2d 229	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴) ↔ (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
19	17, 18	syl5ibcom 154	. . . . . . . 8 ⊢ (𝐴 ⊆ I → (𝑥 = 𝑦 → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
20	9, 19	syl5bi 151	. . . . . . 7 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ I → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
21	20	impd 252	. . . . . 6 ⊢ (𝐴 ⊆ I → ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴) → ⟨𝑥, 𝑦⟩ ∈ 𝐴))
22	6, 21	impbid 128	. . . . 5 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴)))
23	3	opelres 4889	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴))
24	22, 23	bitr4di 197	. . . 4 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴)))
25	24	alrimivv 1863	. . 3 ⊢ (𝐴 ⊆ I → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴)))
26		reli 4733	. . . . 5 ⊢ Rel I
27		relss 4691	. . . . 5 ⊢ (𝐴 ⊆ I → (Rel I → Rel 𝐴))
28	26, 27	mpi 15	. . . 4 ⊢ (𝐴 ⊆ I → Rel 𝐴)
29		relres 4912	. . . 4 ⊢ Rel ( I ↾ dom 𝐴)
30		eqrel 4693	. . . 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 410	. . 3 ⊢ (𝐴 ⊆ I → (𝐴 = ( I ↾ dom 𝐴) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴))))
32	25, 31	mpbird 166	. 2 ⊢ (𝐴 ⊆ I → 𝐴 = ( I ↾ dom 𝐴))
33		resss 4908	. . 3 ⊢ ( I ↾ dom 𝐴) ⊆ I
34		sseq1 3165	. . 3 ⊢ (𝐴 = ( I ↾ dom 𝐴) → (𝐴 ⊆ I ↔ ( I ↾ dom 𝐴) ⊆ I ))
35	33, 34	mpbiri 167	. 2 ⊢ (𝐴 = ( I ↾ dom 𝐴) → 𝐴 ⊆ I )
36	32, 35	impbii 125	1 ⊢ (𝐴 ⊆ I ↔ 𝐴 = ( I ↾ dom 𝐴))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1341   = wceq 1343  ∃wex 1480   ∈ wcel 2136   ⊆ wss 3116  ⟨cop 3579   class class class wbr 3982   I cid 4266  dom cdm 4604   ↾ cres 4606  Rel wrel 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-dm 4614  df-res 4616

This theorem is referenced by:  funcocnv2  5457