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Theorem iss 4679
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 2964 . . . . . . 7 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ I ))
2 vex 2575 . . . . . . . . 9 𝑥 ∈ V
3 vex 2575 . . . . . . . . 9 𝑦 ∈ V
42, 3opeldm 4563 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴)
54a1i 9 . . . . . . 7 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴))
61, 5jcad 295 . . . . . 6 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴)))
7 df-br 3790 . . . . . . . . 9 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
83ideq 4513 . . . . . . . . 9 (𝑥 I 𝑦𝑥 = 𝑦)
97, 8bitr3i 179 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
102eldm2 4558 . . . . . . . . . 10 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
11 opeq2 3575 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
1211eleq1d 2120 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (⟨𝑥, 𝑥⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
1312biimprcd 153 . . . . . . . . . . . . 13 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
149, 13syl5bi 145 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
151, 14sylcom 28 . . . . . . . . . . 11 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
1615exlimdv 1714 . . . . . . . . . 10 (𝐴 ⊆ I → (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
1710, 16syl5bi 145 . . . . . . . . 9 (𝐴 ⊆ I → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
1812imbi2d 223 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴) ↔ (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
1917, 18syl5ibcom 148 . . . . . . . 8 (𝐴 ⊆ I → (𝑥 = 𝑦 → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
209, 19syl5bi 145 . . . . . . 7 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ I → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
2120impd 246 . . . . . 6 (𝐴 ⊆ I → ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴) → ⟨𝑥, 𝑦⟩ ∈ 𝐴))
226, 21impbid 124 . . . . 5 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴)))
233opelres 4642 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴))
2422, 23syl6bbr 191 . . . 4 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴)))
2524alrimivv 1769 . . 3 (𝐴 ⊆ I → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴)))
26 reli 4490 . . . . 5 Rel I
27 relss 4452 . . . . 5 (𝐴 ⊆ I → (Rel I → Rel 𝐴))
2826, 27mpi 15 . . . 4 (𝐴 ⊆ I → Rel 𝐴)
29 relres 4664 . . . 4 Rel ( I ↾ dom 𝐴)
30 eqrel 4454 . . . 4 ((Rel 𝐴 ∧ Rel ( I ↾ dom 𝐴)) → (𝐴 = ( I ↾ dom 𝐴) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴))))
3128, 29, 30sylancl 398 . . 3 (𝐴 ⊆ I → (𝐴 = ( I ↾ dom 𝐴) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴))))
3225, 31mpbird 160 . 2 (𝐴 ⊆ I → 𝐴 = ( I ↾ dom 𝐴))
33 resss 4660 . . 3 ( I ↾ dom 𝐴) ⊆ I
34 sseq1 2991 . . 3 (𝐴 = ( I ↾ dom 𝐴) → (𝐴 ⊆ I ↔ ( I ↾ dom 𝐴) ⊆ I ))
3533, 34mpbiri 161 . 2 (𝐴 = ( I ↾ dom 𝐴) → 𝐴 ⊆ I )
3632, 35impbii 121 1 (𝐴 ⊆ I ↔ 𝐴 = ( I ↾ dom 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1255   = wceq 1257  wex 1395  wcel 1407  wss 2942  cop 3403   class class class wbr 3789   I cid 4050  dom cdm 4370  cres 4372  Rel wrel 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3790  df-opab 3844  df-id 4055  df-xp 4376  df-rel 4377  df-dm 4380  df-res 4382
This theorem is referenced by:  funcocnv2  5176
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