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Theorem sscoid 35542
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 6117 . . 3 Rel ( I ∘ 𝐵)
2 relss 5787 . . 3 (𝐴 ⊆ ( I ∘ 𝐵) → (Rel ( I ∘ 𝐵) → Rel 𝐴))
31, 2mpi 20 . 2 (𝐴 ⊆ ( I ∘ 𝐵) → Rel 𝐴)
4 elrel 5804 . . . . . 6 ((Rel 𝐴𝑥𝐴) → ∃𝑦𝑧 𝑥 = ⟨𝑦, 𝑧⟩)
5 vex 3477 . . . . . . . . . . 11 𝑦 ∈ V
6 vex 3477 . . . . . . . . . . 11 𝑧 ∈ V
75, 6brco 5877 . . . . . . . . . 10 (𝑦( I ∘ 𝐵)𝑧 ↔ ∃𝑥(𝑦𝐵𝑥𝑥 I 𝑧))
86ideq 5859 . . . . . . . . . . . 12 (𝑥 I 𝑧𝑥 = 𝑧)
98anbi1ci 624 . . . . . . . . . . 11 ((𝑦𝐵𝑥𝑥 I 𝑧) ↔ (𝑥 = 𝑧𝑦𝐵𝑥))
109exbii 1842 . . . . . . . . . 10 (∃𝑥(𝑦𝐵𝑥𝑥 I 𝑧) ↔ ∃𝑥(𝑥 = 𝑧𝑦𝐵𝑥))
11 breq2 5156 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑦𝐵𝑥𝑦𝐵𝑧))
1211equsexvw 2000 . . . . . . . . . 10 (∃𝑥(𝑥 = 𝑧𝑦𝐵𝑥) ↔ 𝑦𝐵𝑧)
137, 10, 123bitri 296 . . . . . . . . 9 (𝑦( I ∘ 𝐵)𝑧𝑦𝐵𝑧)
1413a1i 11 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑦( I ∘ 𝐵)𝑧𝑦𝐵𝑧))
15 eleq1 2817 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ ( I ∘ 𝐵)))
16 df-br 5153 . . . . . . . . 9 (𝑦( I ∘ 𝐵)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ( I ∘ 𝐵))
1715, 16bitr4di 288 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑦( I ∘ 𝐵)𝑧))
18 eleq1 2817 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥𝐵 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐵))
19 df-br 5153 . . . . . . . . 9 (𝑦𝐵𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐵)
2018, 19bitr4di 288 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥𝐵𝑦𝐵𝑧))
2114, 17, 203bitr4d 310 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑥𝐵))
2221exlimivv 1927 . . . . . 6 (∃𝑦𝑧 𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑥𝐵))
234, 22syl 17 . . . . 5 ((Rel 𝐴𝑥𝐴) → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑥𝐵))
2423pm5.74da 802 . . . 4 (Rel 𝐴 → ((𝑥𝐴𝑥 ∈ ( I ∘ 𝐵)) ↔ (𝑥𝐴𝑥𝐵)))
2524albidv 1915 . . 3 (Rel 𝐴 → (∀𝑥(𝑥𝐴𝑥 ∈ ( I ∘ 𝐵)) ↔ ∀𝑥(𝑥𝐴𝑥𝐵)))
26 dfss2 3969 . . 3 (𝐴 ⊆ ( I ∘ 𝐵) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ( I ∘ 𝐵)))
27 dfss2 3969 . . 3 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
2825, 26, 273bitr4g 313 . 2 (Rel 𝐴 → (𝐴 ⊆ ( I ∘ 𝐵) ↔ 𝐴𝐵))
293, 28biadanii 820 1 (𝐴 ⊆ ( I ∘ 𝐵) ↔ (Rel 𝐴𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1531   = wceq 1533  wex 1773  wcel 2098  wss 3949  cop 4638   class class class wbr 5152   I cid 5579  ccom 5686  Rel wrel 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-co 5691
This theorem is referenced by:  dffun10  35543
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