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Theorem iss 4597
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.
StepHypRef Expression
1 ssel 2933 . . . . . . 7 (A ⊆ I → (⟨x, y A → ⟨x, y I ))
2 vex 2554 . . . . . . . . 9 x V
3 vex 2554 . . . . . . . . 9 y V
42, 3opeldm 4481 . . . . . . . 8 (⟨x, y Ax dom A)
54a1i 9 . . . . . . 7 (A ⊆ I → (⟨x, y Ax dom A))
61, 5jcad 291 . . . . . 6 (A ⊆ I → (⟨x, y A → (⟨x, y I x dom A)))
7 df-br 3756 . . . . . . . . 9 (x I y ↔ ⟨x, y I )
83ideq 4431 . . . . . . . . 9 (x I yx = y)
97, 8bitr3i 175 . . . . . . . 8 (⟨x, y I ↔ x = y)
102eldm2 4476 . . . . . . . . . 10 (x dom Ayx, y A)
11 opeq2 3541 . . . . . . . . . . . . . . 15 (x = y → ⟨x, x⟩ = ⟨x, y⟩)
1211eleq1d 2103 . . . . . . . . . . . . . 14 (x = y → (⟨x, x A ↔ ⟨x, y A))
1312biimprcd 149 . . . . . . . . . . . . 13 (⟨x, y A → (x = y → ⟨x, x A))
149, 13syl5bi 141 . . . . . . . . . . . 12 (⟨x, y A → (⟨x, y I → ⟨x, x A))
151, 14sylcom 25 . . . . . . . . . . 11 (A ⊆ I → (⟨x, y A → ⟨x, x A))
1615exlimdv 1697 . . . . . . . . . 10 (A ⊆ I → (yx, y A → ⟨x, x A))
1710, 16syl5bi 141 . . . . . . . . 9 (A ⊆ I → (x dom A → ⟨x, x A))
1812imbi2d 219 . . . . . . . . 9 (x = y → ((x dom A → ⟨x, x A) ↔ (x dom A → ⟨x, y A)))
1917, 18syl5ibcom 144 . . . . . . . 8 (A ⊆ I → (x = y → (x dom A → ⟨x, y A)))
209, 19syl5bi 141 . . . . . . 7 (A ⊆ I → (⟨x, y I → (x dom A → ⟨x, y A)))
2120impd 242 . . . . . 6 (A ⊆ I → ((⟨x, y I x dom A) → ⟨x, y A))
226, 21impbid 120 . . . . 5 (A ⊆ I → (⟨x, y A ↔ (⟨x, y I x dom A)))
233opelres 4560 . . . . 5 (⟨x, y ( I ↾ dom A) ↔ (⟨x, y I x dom A))
2422, 23syl6bbr 187 . . . 4 (A ⊆ I → (⟨x, y A ↔ ⟨x, y ( I ↾ dom A)))
2524alrimivv 1752 . . 3 (A ⊆ I → xy(⟨x, y A ↔ ⟨x, y ( I ↾ dom A)))
26 reli 4408 . . . . 5 Rel I
27 relss 4370 . . . . 5 (A ⊆ I → (Rel I → Rel A))
2826, 27mpi 15 . . . 4 (A ⊆ I → Rel A)
29 relres 4582 . . . 4 Rel ( I ↾ dom A)
30 eqrel 4372 . . . 4 ((Rel A Rel ( I ↾ dom A)) → (A = ( I ↾ dom A) ↔ xy(⟨x, y A ↔ ⟨x, y ( I ↾ dom A))))
3128, 29, 30sylancl 392 . . 3 (A ⊆ I → (A = ( I ↾ dom A) ↔ xy(⟨x, y A ↔ ⟨x, y ( I ↾ dom A))))
3225, 31mpbird 156 . 2 (A ⊆ I → A = ( I ↾ dom A))
33 resss 4578 . . 3 ( I ↾ dom A) ⊆ I
34 sseq1 2960 . . 3 (A = ( I ↾ dom A) → (A ⊆ I ↔ ( I ↾ dom A) ⊆ I ))
3533, 34mpbiri 157 . 2 (A = ( I ↾ dom A) → A ⊆ I )
3632, 35impbii 117 1 (A ⊆ I ↔ A = ( I ↾ dom A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242  wex 1378   wcel 1390  wss 2911  cop 3370   class class class wbr 3755   I cid 4016  dom cdm 4288  cres 4290  Rel wrel 4293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-dm 4298  df-res 4300
This theorem is referenced by:  funcocnv2  5094
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