Theorem iss 4955

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 3151	. . . . . . 7 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ I ))
2		vex 2742	. . . . . . . . 9 ⊢ 𝑥 ∈ V
3		vex 2742	. . . . . . . . 9 ⊢ 𝑦 ∈ V
4	2, 3	opeldm 4832	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
5	4	a1i 9	. . . . . . 7 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴))
6	1, 5	jcad 307	. . . . . 6 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴)))
7		df-br 4006	. . . . . . . . 9 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
8	3	ideq 4781	. . . . . . . . 9 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
9	7, 8	bitr3i 186	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
10	2	eldm2 4827	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
11		opeq2 3781	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
12	11	eleq1d 2246	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (⟨𝑥, 𝑥⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
13	12	biimprcd 160	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
14	9, 13	biimtrid 152	. . . . . . . . . . . 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 1819	. . . . . . . . . 10 ⊢ (𝐴 ⊆ I → (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
17	10, 16	biimtrid 152	. . . . . . . . 9 ⊢ (𝐴 ⊆ I → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
18	12	imbi2d 230	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴) ↔ (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
19	17, 18	syl5ibcom 155	. . . . . . . 8 ⊢ (𝐴 ⊆ I → (𝑥 = 𝑦 → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
20	9, 19	biimtrid 152	. . . . . . 7 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ I → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
21	20	impd 254	. . . . . 6 ⊢ (𝐴 ⊆ I → ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴) → ⟨𝑥, 𝑦⟩ ∈ 𝐴))
22	6, 21	impbid 129	. . . . 5 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴)))
23	3	opelres 4914	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴))
24	22, 23	bitr4di 198	. . . 4 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴)))
25	24	alrimivv 1875	. . 3 ⊢ (𝐴 ⊆ I → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴)))
26		reli 4758	. . . . 5 ⊢ Rel I
27		relss 4715	. . . . 5 ⊢ (𝐴 ⊆ I → (Rel I → Rel 𝐴))
28	26, 27	mpi 15	. . . 4 ⊢ (𝐴 ⊆ I → Rel 𝐴)
29		relres 4937	. . . 4 ⊢ Rel ( I ↾ dom 𝐴)
30		eqrel 4717	. . . 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 413	. . 3 ⊢ (𝐴 ⊆ I → (𝐴 = ( I ↾ dom 𝐴) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴))))
32	25, 31	mpbird 167	. 2 ⊢ (𝐴 ⊆ I → 𝐴 = ( I ↾ dom 𝐴))
33		resss 4933	. . 3 ⊢ ( I ↾ dom 𝐴) ⊆ I
34		sseq1 3180	. . 3 ⊢ (𝐴 = ( I ↾ dom 𝐴) → (𝐴 ⊆ I ↔ ( I ↾ dom 𝐴) ⊆ I ))
35	33, 34	mpbiri 168	. 2 ⊢ (𝐴 = ( I ↾ dom 𝐴) → 𝐴 ⊆ I )
36	32, 35	impbii 126	1 ⊢ (𝐴 ⊆ I ↔ 𝐴 = ( I ↾ dom 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1351   = wceq 1353  ∃wex 1492   ∈ wcel 2148   ⊆ wss 3131  ⟨cop 3597   class class class wbr 4005   I cid 4290  dom cdm 4628   ↾ cres 4630  Rel wrel 4633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-dm 4638  df-res 4640

This theorem is referenced by:  funcocnv2  5488