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Theorem sscoid 32485
Description: A condition for subset and composition with identity. (Contributed by Scott Fenton, 13-Apr-2018.)
Assertion
Ref Expression
sscoid (𝐴 ⊆ ( I ∘ 𝐵) ↔ (Rel 𝐴𝐴𝐵))

Proof of Theorem sscoid
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relco 5821 . . 3 Rel ( I ∘ 𝐵)
2 relss 5378 . . 3 (𝐴 ⊆ ( I ∘ 𝐵) → (Rel ( I ∘ 𝐵) → Rel 𝐴))
31, 2mpi 20 . 2 (𝐴 ⊆ ( I ∘ 𝐵) → Rel 𝐴)
4 elrel 5393 . . . . . 6 ((Rel 𝐴𝑥𝐴) → ∃𝑦𝑧 𝑥 = ⟨𝑦, 𝑧⟩)
5 vex 3353 . . . . . . . . . . 11 𝑦 ∈ V
6 vex 3353 . . . . . . . . . . 11 𝑧 ∈ V
75, 6brco 5463 . . . . . . . . . 10 (𝑦( I ∘ 𝐵)𝑧 ↔ ∃𝑥(𝑦𝐵𝑥𝑥 I 𝑧))
8 ancom 452 . . . . . . . . . . . . 13 ((𝑦𝐵𝑥𝑥 I 𝑧) ↔ (𝑥 I 𝑧𝑦𝐵𝑥))
96ideq 5445 . . . . . . . . . . . . . 14 (𝑥 I 𝑧𝑥 = 𝑧)
109anbi1i 617 . . . . . . . . . . . . 13 ((𝑥 I 𝑧𝑦𝐵𝑥) ↔ (𝑥 = 𝑧𝑦𝐵𝑥))
118, 10bitri 266 . . . . . . . . . . . 12 ((𝑦𝐵𝑥𝑥 I 𝑧) ↔ (𝑥 = 𝑧𝑦𝐵𝑥))
1211exbii 1943 . . . . . . . . . . 11 (∃𝑥(𝑦𝐵𝑥𝑥 I 𝑧) ↔ ∃𝑥(𝑥 = 𝑧𝑦𝐵𝑥))
13 breq2 4815 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑦𝐵𝑥𝑦𝐵𝑧))
146, 13ceqsexv 3395 . . . . . . . . . . 11 (∃𝑥(𝑥 = 𝑧𝑦𝐵𝑥) ↔ 𝑦𝐵𝑧)
1512, 14bitri 266 . . . . . . . . . 10 (∃𝑥(𝑦𝐵𝑥𝑥 I 𝑧) ↔ 𝑦𝐵𝑧)
167, 15bitri 266 . . . . . . . . 9 (𝑦( I ∘ 𝐵)𝑧𝑦𝐵𝑧)
1716a1i 11 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑦( I ∘ 𝐵)𝑧𝑦𝐵𝑧))
18 eleq1 2832 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ ( I ∘ 𝐵)))
19 df-br 4812 . . . . . . . . 9 (𝑦( I ∘ 𝐵)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ( I ∘ 𝐵))
2018, 19syl6bbr 280 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑦( I ∘ 𝐵)𝑧))
21 eleq1 2832 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥𝐵 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐵))
22 df-br 4812 . . . . . . . . 9 (𝑦𝐵𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐵)
2321, 22syl6bbr 280 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥𝐵𝑦𝐵𝑧))
2417, 20, 233bitr4d 302 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑥𝐵))
2524exlimivv 2027 . . . . . 6 (∃𝑦𝑧 𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑥𝐵))
264, 25syl 17 . . . . 5 ((Rel 𝐴𝑥𝐴) → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑥𝐵))
2726pm5.74da 838 . . . 4 (Rel 𝐴 → ((𝑥𝐴𝑥 ∈ ( I ∘ 𝐵)) ↔ (𝑥𝐴𝑥𝐵)))
2827albidv 2015 . . 3 (Rel 𝐴 → (∀𝑥(𝑥𝐴𝑥 ∈ ( I ∘ 𝐵)) ↔ ∀𝑥(𝑥𝐴𝑥𝐵)))
29 dfss2 3751 . . 3 (𝐴 ⊆ ( I ∘ 𝐵) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ( I ∘ 𝐵)))
30 dfss2 3751 . . 3 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
3128, 29, 303bitr4g 305 . 2 (Rel 𝐴 → (𝐴 ⊆ ( I ∘ 𝐵) ↔ 𝐴𝐵))
323, 31biadan2 853 1 (𝐴 ⊆ ( I ∘ 𝐵) ↔ (Rel 𝐴𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650   = wceq 1652  wex 1874  wcel 2155  wss 3734  cop 4342   class class class wbr 4811   I cid 5186  ccom 5283  Rel wrel 5284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-br 4812  df-opab 4874  df-id 5187  df-xp 5285  df-rel 5286  df-co 5288
This theorem is referenced by:  dffun10  32486
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