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Theorem sscoid 32357
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 5777 . . 3 Rel ( I ∘ 𝐵)
2 relss 5346 . . 3 (𝐴 ⊆ ( I ∘ 𝐵) → (Rel ( I ∘ 𝐵) → Rel 𝐴))
31, 2mpi 20 . 2 (𝐴 ⊆ ( I ∘ 𝐵) → Rel 𝐴)
4 elrel 5362 . . . . . 6 ((Rel 𝐴𝑥𝐴) → ∃𝑦𝑧 𝑥 = ⟨𝑦, 𝑧⟩)
5 vex 3354 . . . . . . . . . . 11 𝑦 ∈ V
6 vex 3354 . . . . . . . . . . 11 𝑧 ∈ V
75, 6brco 5431 . . . . . . . . . 10 (𝑦( I ∘ 𝐵)𝑧 ↔ ∃𝑥(𝑦𝐵𝑥𝑥 I 𝑧))
8 ancom 452 . . . . . . . . . . . . 13 ((𝑦𝐵𝑥𝑥 I 𝑧) ↔ (𝑥 I 𝑧𝑦𝐵𝑥))
96ideq 5413 . . . . . . . . . . . . . 14 (𝑥 I 𝑧𝑥 = 𝑧)
109anbi1i 602 . . . . . . . . . . . . 13 ((𝑥 I 𝑧𝑦𝐵𝑥) ↔ (𝑥 = 𝑧𝑦𝐵𝑥))
118, 10bitri 264 . . . . . . . . . . . 12 ((𝑦𝐵𝑥𝑥 I 𝑧) ↔ (𝑥 = 𝑧𝑦𝐵𝑥))
1211exbii 1924 . . . . . . . . . . 11 (∃𝑥(𝑦𝐵𝑥𝑥 I 𝑧) ↔ ∃𝑥(𝑥 = 𝑧𝑦𝐵𝑥))
13 breq2 4790 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑦𝐵𝑥𝑦𝐵𝑧))
146, 13ceqsexv 3393 . . . . . . . . . . 11 (∃𝑥(𝑥 = 𝑧𝑦𝐵𝑥) ↔ 𝑦𝐵𝑧)
1512, 14bitri 264 . . . . . . . . . 10 (∃𝑥(𝑦𝐵𝑥𝑥 I 𝑧) ↔ 𝑦𝐵𝑧)
167, 15bitri 264 . . . . . . . . 9 (𝑦( I ∘ 𝐵)𝑧𝑦𝐵𝑧)
1716a1i 11 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑦( I ∘ 𝐵)𝑧𝑦𝐵𝑧))
18 eleq1 2838 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ ( I ∘ 𝐵)))
19 df-br 4787 . . . . . . . . 9 (𝑦( I ∘ 𝐵)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ( I ∘ 𝐵))
2018, 19syl6bbr 278 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑦( I ∘ 𝐵)𝑧))
21 eleq1 2838 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥𝐵 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐵))
22 df-br 4787 . . . . . . . . 9 (𝑦𝐵𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐵)
2321, 22syl6bbr 278 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥𝐵𝑦𝐵𝑧))
2417, 20, 233bitr4d 300 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑥𝐵))
2524exlimivv 2012 . . . . . 6 (∃𝑦𝑧 𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑥𝐵))
264, 25syl 17 . . . . 5 ((Rel 𝐴𝑥𝐴) → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑥𝐵))
2726pm5.74da 817 . . . 4 (Rel 𝐴 → ((𝑥𝐴𝑥 ∈ ( I ∘ 𝐵)) ↔ (𝑥𝐴𝑥𝐵)))
2827albidv 2001 . . 3 (Rel 𝐴 → (∀𝑥(𝑥𝐴𝑥 ∈ ( I ∘ 𝐵)) ↔ ∀𝑥(𝑥𝐴𝑥𝐵)))
29 dfss2 3740 . . 3 (𝐴 ⊆ ( I ∘ 𝐵) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ( I ∘ 𝐵)))
30 dfss2 3740 . . 3 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
3128, 29, 303bitr4g 303 . 2 (Rel 𝐴 → (𝐴 ⊆ ( I ∘ 𝐵) ↔ 𝐴𝐵))
323, 31biadan2 802 1 (𝐴 ⊆ ( I ∘ 𝐵) ↔ (Rel 𝐴𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1629   = wceq 1631  wex 1852  wcel 2145  wss 3723  cop 4322   class class class wbr 4786   I cid 5156  ccom 5253  Rel wrel 5254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-co 5258
This theorem is referenced by:  dffun10  32358
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