Theorem iss 2599

Description: A subclass of the identity function is the identity function restricted to its domain.

Assertion

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

Proof of Theorem iss

Step	Hyp	Ref	Expression
1		ssel 1502	. . . . . . . 8 ⊢ (A ⊆ I → (⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ I))
2		df-br 2063	. . . . . . . . 9 ⊢ (xIy ↔ ⟨x, y⟩ ∈ I)
3		visset 1350	. . . . . . . . . 10 ⊢ x ∈ V
4		visset 1350	. . . . . . . . . 10 ⊢ y ∈ V
5	3, 4	ideq 2127	. . . . . . . . 9 ⊢ (xIy ↔ x = y)
6	2, 5	bitr3 153	. . . . . . . 8 ⊢ (⟨x, y⟩ ∈ I ↔ x = y)
7	1, 6	syl6ib 185	. . . . . . 7 ⊢ (A ⊆ I → (⟨x, y⟩ ∈ A → x = y))
8		pm4.71r 482	. . . . . . 7 ⊢ ((⟨x, y⟩ ∈ A → x = y) ↔ (⟨x, y⟩ ∈ A ↔ (x = y ∧ ⟨x, y⟩ ∈ A)))
9	7, 8	sylib 173	. . . . . 6 ⊢ (A ⊆ I → (⟨x, y⟩ ∈ A ↔ (x = y ∧ ⟨x, y⟩ ∈ A)))
10		cleqcom 1103	. . . . . . . . . . . . 13 ⊢ (x = y ↔ y = x)
11	10	anbi1i 368	. . . . . . . . . . . 12 ⊢ ((x = y ∧ ⟨x, y⟩ ∈ A) ↔ (y = x ∧ ⟨x, y⟩ ∈ A))
12	9, 11	syl6bb 414	. . . . . . . . . . 11 ⊢ (A ⊆ I → (⟨x, y⟩ ∈ A ↔ (y = x ∧ ⟨x, y⟩ ∈ A)))
13	12	biexdv 936	. . . . . . . . . 10 ⊢ (A ⊆ I → (∃y⟨x, y⟩ ∈ A ↔ ∃y(y = x ∧ ⟨x, y⟩ ∈ A)))
14		opeq2 1877	. . . . . . . . . . . 12 ⊢ (y = x → ⟨x, y⟩ = ⟨x, x⟩)
15	14	eleq1d 1155	. . . . . . . . . . 11 ⊢ (y = x → (⟨x, y⟩ ∈ A ↔ ⟨x, x⟩ ∈ A))
16	3, 15	ceqsexv 1371	. . . . . . . . . 10 ⊢ (∃y(y = x ∧ ⟨x, y⟩ ∈ A) ↔ ⟨x, x⟩ ∈ A)
17	13, 16	syl6bb 414	. . . . . . . . 9 ⊢ (A ⊆ I → (∃y⟨x, y⟩ ∈ A ↔ ⟨x, x⟩ ∈ A))
18	3	eldm2 2528	. . . . . . . . 9 ⊢ (x ∈ dom A ↔ ∃y⟨x, y⟩ ∈ A)
19	17, 18	syl5bb 410	. . . . . . . 8 ⊢ (A ⊆ I → (x ∈ dom A ↔ ⟨x, x⟩ ∈ A))
20	19	anbi2d 468	. . . . . . 7 ⊢ (A ⊆ I → ((x = y ∧ x ∈ dom A) ↔ (x = y ∧ ⟨x, x⟩ ∈ A)))
21		opeq2 1877	. . . . . . . . 9 ⊢ (x = y → ⟨x, x⟩ = ⟨x, y⟩)
22	21	eleq1d 1155	. . . . . . . 8 ⊢ (x = y → (⟨x, x⟩ ∈ A ↔ ⟨x, y⟩ ∈ A))
23	22	pm5.32i 489	. . . . . . 7 ⊢ ((x = y ∧ ⟨x, x⟩ ∈ A) ↔ (x = y ∧ ⟨x, y⟩ ∈ A))
24	20, 23	syl6bb 414	. . . . . 6 ⊢ (A ⊆ I → ((x = y ∧ x ∈ dom A) ↔ (x = y ∧ ⟨x, y⟩ ∈ A)))
25	9, 24	bitr4d 409	. . . . 5 ⊢ (A ⊆ I → (⟨x, y⟩ ∈ A ↔ (x = y ∧ x ∈ dom A)))
26	4	opelres 2579	. . . . . 6 ⊢ (⟨x, y⟩ ∈ (I ↾ dom A) ↔ (⟨x, y⟩ ∈ I ∧ x ∈ dom A))
27	6	anbi1i 368	. . . . . 6 ⊢ ((⟨x, y⟩ ∈ I ∧ x ∈ dom A) ↔ (x = y ∧ x ∈ dom A))
28	26, 27	bitr2 152	. . . . 5 ⊢ ((x = y ∧ x ∈ dom A) ↔ ⟨x, y⟩ ∈ (I ↾ dom A))
29	25, 28	syl6bb 414	. . . 4 ⊢ (A ⊆ I → (⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ (I ↾ dom A)))
30	29	19.21aivv 944	. . 3 ⊢ (A ⊆ I → ∀x∀y(⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ (I ↾ dom A)))
31		reli 2500	. . . . 5 ⊢ Rel I
32		ssrel 2479	. . . . 5 ⊢ (A ⊆ I → (Rel I → Rel A))
33	31, 32	mpi 44	. . . 4 ⊢ (A ⊆ I → Rel A)
34		relres 2591	. . . . 5 ⊢ Rel (I ↾ dom A)
35		cleqrel 2483	. . . . 5 ⊢ ((Rel A ∧ Rel (I ↾ dom A)) → (A = (I ↾ dom A) ↔ ∀x∀y(⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ (I ↾ dom A))))
36	34, 35	mpan2 519	. . . 4 ⊢ (Rel A → (A = (I ↾ dom A) ↔ ∀x∀y(⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ (I ↾ dom A))))
37	33, 36	syl 12	. . 3 ⊢ (A ⊆ I → (A = (I ↾ dom A) ↔ ∀x∀y(⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ (I ↾ dom A))))
38	30, 37	mpbird 171	. 2 ⊢ (A ⊆ I → A = (I ↾ dom A))
39		resss 2587	. . 3 ⊢ (I ↾ dom A) ⊆ I
40		sseq1 1521	. . 3 ⊢ (A = (I ↾ dom A) → (A ⊆ I ↔ (I ↾ dom A) ⊆ I))
41	39, 40	mpbiri 169	. 2 ⊢ (A = (I ↾ dom A) → A ⊆ I)
42	38, 41	impbi 139	1 ⊢ (A ⊆ I ↔ A = (I ↾ dom A))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487  ⟨cop 1810   class class class wbr 2054  Icid 2057  dom cdm 2410   ↾ cres 2412  Rel wrel 2415

This theorem is referenced by:  f1ococnv2 2817

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-dm 2428  df-res 2430

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