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Theorem sscoid 35914
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 6126 . . 3 Rel ( I ∘ 𝐵)
2 relss 5791 . . 3 (𝐴 ⊆ ( I ∘ 𝐵) → (Rel ( I ∘ 𝐵) → Rel 𝐴))
31, 2mpi 20 . 2 (𝐴 ⊆ ( I ∘ 𝐵) → Rel 𝐴)
4 elrel 5808 . . . . . 6 ((Rel 𝐴𝑥𝐴) → ∃𝑦𝑧 𝑥 = ⟨𝑦, 𝑧⟩)
5 vex 3484 . . . . . . . . . . 11 𝑦 ∈ V
6 vex 3484 . . . . . . . . . . 11 𝑧 ∈ V
75, 6brco 5881 . . . . . . . . . 10 (𝑦( I ∘ 𝐵)𝑧 ↔ ∃𝑥(𝑦𝐵𝑥𝑥 I 𝑧))
86ideq 5863 . . . . . . . . . . . 12 (𝑥 I 𝑧𝑥 = 𝑧)
98anbi1ci 626 . . . . . . . . . . 11 ((𝑦𝐵𝑥𝑥 I 𝑧) ↔ (𝑥 = 𝑧𝑦𝐵𝑥))
109exbii 1848 . . . . . . . . . 10 (∃𝑥(𝑦𝐵𝑥𝑥 I 𝑧) ↔ ∃𝑥(𝑥 = 𝑧𝑦𝐵𝑥))
11 breq2 5147 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑦𝐵𝑥𝑦𝐵𝑧))
1211equsexvw 2004 . . . . . . . . . 10 (∃𝑥(𝑥 = 𝑧𝑦𝐵𝑥) ↔ 𝑦𝐵𝑧)
137, 10, 123bitri 297 . . . . . . . . 9 (𝑦( I ∘ 𝐵)𝑧𝑦𝐵𝑧)
1413a1i 11 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑦( I ∘ 𝐵)𝑧𝑦𝐵𝑧))
15 eleq1 2829 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ ( I ∘ 𝐵)))
16 df-br 5144 . . . . . . . . 9 (𝑦( I ∘ 𝐵)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ( I ∘ 𝐵))
1715, 16bitr4di 289 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑦( I ∘ 𝐵)𝑧))
18 eleq1 2829 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥𝐵 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐵))
19 df-br 5144 . . . . . . . . 9 (𝑦𝐵𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐵)
2018, 19bitr4di 289 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥𝐵𝑦𝐵𝑧))
2114, 17, 203bitr4d 311 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑥𝐵))
2221exlimivv 1932 . . . . . 6 (∃𝑦𝑧 𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑥𝐵))
234, 22syl 17 . . . . 5 ((Rel 𝐴𝑥𝐴) → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑥𝐵))
2423pm5.74da 804 . . . 4 (Rel 𝐴 → ((𝑥𝐴𝑥 ∈ ( I ∘ 𝐵)) ↔ (𝑥𝐴𝑥𝐵)))
2524albidv 1920 . . 3 (Rel 𝐴 → (∀𝑥(𝑥𝐴𝑥 ∈ ( I ∘ 𝐵)) ↔ ∀𝑥(𝑥𝐴𝑥𝐵)))
26 df-ss 3968 . . 3 (𝐴 ⊆ ( I ∘ 𝐵) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ( I ∘ 𝐵)))
27 df-ss 3968 . . 3 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
2825, 26, 273bitr4g 314 . 2 (Rel 𝐴 → (𝐴 ⊆ ( I ∘ 𝐵) ↔ 𝐴𝐵))
293, 28biadanii 822 1 (𝐴 ⊆ ( I ∘ 𝐵) ↔ (Rel 𝐴𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wex 1779  wcel 2108  wss 3951  cop 4632   class class class wbr 5143   I cid 5577  ccom 5689  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-co 5694
This theorem is referenced by:  dffun10  35915
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