Theorem iss 5001

Description: A subclass of the identity function is the identity function restricted to its domain. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 13-Dec-2003.) (Revised by set.mm contributors, 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 3268	. . . . . 6 ⊢ (A ⊆ I → (⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ I ))
2		opeldm 4911	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ A → x ∈ dom A)
3	2	a1i 10	. . . . . 6 ⊢ (A ⊆ I → (⟨x, y⟩ ∈ A → x ∈ dom A))
4	1, 3	jcad 519	. . . . 5 ⊢ (A ⊆ I → (⟨x, y⟩ ∈ A → (⟨x, y⟩ ∈ I ∧ x ∈ dom A)))
5		df-br 4641	. . . . . . . 8 ⊢ (x I y ↔ ⟨x, y⟩ ∈ I )
6		vex 2863	. . . . . . . . 9 ⊢ y ∈ V
7	6	ideq 4871	. . . . . . . 8 ⊢ (x I y ↔ x = y)
8	5, 7	bitr3i 242	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ I ↔ x = y)
9	8	anbi1i 676	. . . . . 6 ⊢ ((⟨x, y⟩ ∈ I ∧ x ∈ dom A) ↔ (x = y ∧ x ∈ dom A))
10		eldm2 4900	. . . . . . . . 9 ⊢ (x ∈ dom A ↔ ∃y⟨x, y⟩ ∈ A)
11	1, 8	syl6ib 217	. . . . . . . . . . 11 ⊢ (A ⊆ I → (⟨x, y⟩ ∈ A → x = y))
12		opeq2 4580	. . . . . . . . . . . . 13 ⊢ (x = y → ⟨x, x⟩ = ⟨x, y⟩)
13	12	eleq1d 2419	. . . . . . . . . . . 12 ⊢ (x = y → (⟨x, x⟩ ∈ A ↔ ⟨x, y⟩ ∈ A))
14	13	biimprd 214	. . . . . . . . . . 11 ⊢ (x = y → (⟨x, y⟩ ∈ A → ⟨x, x⟩ ∈ A))
15	11, 14	syli 33	. . . . . . . . . 10 ⊢ (A ⊆ I → (⟨x, y⟩ ∈ A → ⟨x, x⟩ ∈ A))
16	15	exlimdv 1636	. . . . . . . . 9 ⊢ (A ⊆ I → (∃y⟨x, y⟩ ∈ A → ⟨x, x⟩ ∈ A))
17	10, 16	syl5bi 208	. . . . . . . 8 ⊢ (A ⊆ I → (x ∈ dom A → ⟨x, x⟩ ∈ A))
18	13	biimpd 198	. . . . . . . 8 ⊢ (x = y → (⟨x, x⟩ ∈ A → ⟨x, y⟩ ∈ A))
19	17, 18	syl9 66	. . . . . . 7 ⊢ (A ⊆ I → (x = y → (x ∈ dom A → ⟨x, y⟩ ∈ A)))
20	19	imp3a 420	. . . . . 6 ⊢ (A ⊆ I → ((x = y ∧ x ∈ dom A) → ⟨x, y⟩ ∈ A))
21	9, 20	syl5bi 208	. . . . 5 ⊢ (A ⊆ I → ((⟨x, y⟩ ∈ I ∧ x ∈ dom A) → ⟨x, y⟩ ∈ A))
22	4, 21	impbid 183	. . . 4 ⊢ (A ⊆ I → (⟨x, y⟩ ∈ A ↔ (⟨x, y⟩ ∈ I ∧ x ∈ dom A)))
23		opelres 4951	. . . 4 ⊢ (⟨x, y⟩ ∈ ( I ↾ dom A) ↔ (⟨x, y⟩ ∈ I ∧ x ∈ dom A))
24	22, 23	syl6bbr 254	. . 3 ⊢ (A ⊆ I → (⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ ( I ↾ dom A)))
25	24	eqrelrdv 4853	. 2 ⊢ (A ⊆ I → A = ( I ↾ dom A))
26		resss 4989	. . 3 ⊢ ( I ↾ dom A) ⊆ I
27		sseq1 3293	. . 3 ⊢ (A = ( I ↾ dom A) → (A ⊆ I ↔ ( I ↾ dom A) ⊆ I ))
28	26, 27	mpbiri 224	. 2 ⊢ (A = ( I ↾ dom A) → A ⊆ I )
29	25, 28	impbii 180	1 ⊢ (A ⊆ I ↔ A = ( I ↾ dom A))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ⊆ wss 3258  ⟨cop 4562   class class class wbr 4640   I cid 4764  dom cdm 4773   ↾ cres 4775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789

This theorem is referenced by:  f1ococnv2  5310