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Theorem iss 5367
 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 3562 . . . . . . 7 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ I ))
2 vex 3176 . . . . . . . . 9 𝑥 ∈ V
3 vex 3176 . . . . . . . . 9 𝑦 ∈ V
42, 3opeldm 5250 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴)
54a1i 11 . . . . . . 7 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴))
61, 5jcad 554 . . . . . 6 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴)))
7 df-br 4584 . . . . . . . . 9 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
83ideq 5196 . . . . . . . . 9 (𝑥 I 𝑦𝑥 = 𝑦)
97, 8bitr3i 265 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
102eldm2 5244 . . . . . . . . . 10 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
11 opeq2 4341 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
1211eleq1d 2672 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (⟨𝑥, 𝑥⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
1312biimprcd 239 . . . . . . . . . . . . 13 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
149, 13syl5bi 231 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
151, 14sylcom 30 . . . . . . . . . . 11 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
1615exlimdv 1848 . . . . . . . . . 10 (𝐴 ⊆ I → (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
1710, 16syl5bi 231 . . . . . . . . 9 (𝐴 ⊆ I → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
1812imbi2d 329 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴) ↔ (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
1917, 18syl5ibcom 234 . . . . . . . 8 (𝐴 ⊆ I → (𝑥 = 𝑦 → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
209, 19syl5bi 231 . . . . . . 7 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ I → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
2120impd 446 . . . . . 6 (𝐴 ⊆ I → ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴) → ⟨𝑥, 𝑦⟩ ∈ 𝐴))
226, 21impbid 201 . . . . 5 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴)))
233opelres 5322 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴))
2422, 23syl6bbr 277 . . . 4 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴)))
2524alrimivv 1843 . . 3 (𝐴 ⊆ I → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴)))
26 reli 5171 . . . . 5 Rel I
27 relss 5129 . . . . 5 (𝐴 ⊆ I → (Rel I → Rel 𝐴))
2826, 27mpi 20 . . . 4 (𝐴 ⊆ I → Rel 𝐴)
29 relres 5346 . . . 4 Rel ( I ↾ dom 𝐴)
30 eqrel 5132 . . . 4 ((Rel 𝐴 ∧ Rel ( I ↾ dom 𝐴)) → (𝐴 = ( I ↾ dom 𝐴) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴))))
3128, 29, 30sylancl 693 . . 3 (𝐴 ⊆ I → (𝐴 = ( I ↾ dom 𝐴) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴))))
3225, 31mpbird 246 . 2 (𝐴 ⊆ I → 𝐴 = ( I ↾ dom 𝐴))
33 resss 5342 . . 3 ( I ↾ dom 𝐴) ⊆ I
34 sseq1 3589 . . 3 (𝐴 = ( I ↾ dom 𝐴) → (𝐴 ⊆ I ↔ ( I ↾ dom 𝐴) ⊆ I ))
3533, 34mpbiri 247 . 2 (𝐴 = ( I ↾ dom 𝐴) → 𝐴 ⊆ I )
3632, 35impbii 198 1 (𝐴 ⊆ I ↔ 𝐴 = ( I ↾ dom 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ⊆ wss 3540  ⟨cop 4131   class class class wbr 4583   I cid 4948  dom cdm 5038   ↾ cres 5040  Rel wrel 5043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-dm 5048  df-res 5050 This theorem is referenced by:  funcocnv2  6074  trust  21843
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