Theorem iss 5905

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 3499	. . . . . . . . 9 ⊢ 𝑥 ∈ V
2		vex 3499	. . . . . . . . 9 ⊢ 𝑦 ∈ V
3	1, 2	opeldm 5778	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
4	3	a1i 11	. . . . . . 7 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴))
5		ssel 3963	. . . . . . 7 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ I ))
6	4, 5	jcad 515	. . . . . 6 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I )))
7		df-br 5069	. . . . . . . . 9 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
8	2	ideq 5725	. . . . . . . . 9 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
9	7, 8	bitr3i 279	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
10	1	eldm2 5772	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
11		opeq2 4806	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
12	11	eleq1d 2899	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (⟨𝑥, 𝑥⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
13	12	biimprcd 252	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
14	9, 13	syl5bi 244	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
15	5, 14	sylcom 30	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
16	15	exlimdv 1934	. . . . . . . . . 10 ⊢ (𝐴 ⊆ I → (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
17	10, 16	syl5bi 244	. . . . . . . . 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 247	. . . . . . . 8 ⊢ (𝐴 ⊆ I → (𝑥 = 𝑦 → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
20	9, 19	syl5bi 244	. . . . . . 7 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ I → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
21	20	impcomd 414	. . . . . 6 ⊢ (𝐴 ⊆ I → ((𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) → ⟨𝑥, 𝑦⟩ ∈ 𝐴))
22	6, 21	impbid 214	. . . . 5 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I )))
23	2	opelresi 5863	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴) ↔ (𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
24	22, 23	syl6bbr 291	. . . 4 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴)))
25	24	alrimivv 1929	. . 3 ⊢ (𝐴 ⊆ I → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴)))
26		reli 5700	. . . . 5 ⊢ Rel I
27		relss 5658	. . . . 5 ⊢ (𝐴 ⊆ I → (Rel I → Rel 𝐴))
28	26, 27	mpi 20	. . . 4 ⊢ (𝐴 ⊆ I → Rel 𝐴)
29		relres 5884	. . . 4 ⊢ Rel ( I ↾ dom 𝐴)
30		eqrel 5660	. . . 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 588	. . 3 ⊢ (𝐴 ⊆ I → (𝐴 = ( I ↾ dom 𝐴) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴))))
32	25, 31	mpbird 259	. 2 ⊢ (𝐴 ⊆ I → 𝐴 = ( I ↾ dom 𝐴))
33		resss 5880	. . 3 ⊢ ( I ↾ dom 𝐴) ⊆ I
34		sseq1 3994	. . 3 ⊢ (𝐴 = ( I ↾ dom 𝐴) → (𝐴 ⊆ I ↔ ( I ↾ dom 𝐴) ⊆ I ))
35	33, 34	mpbiri 260	. 2 ⊢ (𝐴 = ( I ↾ dom 𝐴) → 𝐴 ⊆ I )
36	32, 35	impbii 211	1 ⊢ (𝐴 ⊆ I ↔ 𝐴 = ( I ↾ dom 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ⊆ wss 3938  ⟨cop 4575   class class class wbr 5068   I cid 5461  dom cdm 5557   ↾ cres 5559  Rel wrel 5562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-dm 5567  df-res 5569

This theorem is referenced by:  funcocnv2  6641  trust  22840