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Theorem sscoid 32983
 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 5972 . . 3 Rel ( I ∘ 𝐵)
2 relss 5542 . . 3 (𝐴 ⊆ ( I ∘ 𝐵) → (Rel ( I ∘ 𝐵) → Rel 𝐴))
31, 2mpi 20 . 2 (𝐴 ⊆ ( I ∘ 𝐵) → Rel 𝐴)
4 elrel 5557 . . . . . 6 ((Rel 𝐴𝑥𝐴) → ∃𝑦𝑧 𝑥 = ⟨𝑦, 𝑧⟩)
5 vex 3440 . . . . . . . . . . 11 𝑦 ∈ V
6 vex 3440 . . . . . . . . . . 11 𝑧 ∈ V
75, 6brco 5627 . . . . . . . . . 10 (𝑦( I ∘ 𝐵)𝑧 ↔ ∃𝑥(𝑦𝐵𝑥𝑥 I 𝑧))
86ideq 5609 . . . . . . . . . . . 12 (𝑥 I 𝑧𝑥 = 𝑧)
98anbi1ci 625 . . . . . . . . . . 11 ((𝑦𝐵𝑥𝑥 I 𝑧) ↔ (𝑥 = 𝑧𝑦𝐵𝑥))
109exbii 1829 . . . . . . . . . 10 (∃𝑥(𝑦𝐵𝑥𝑥 I 𝑧) ↔ ∃𝑥(𝑥 = 𝑧𝑦𝐵𝑥))
11 breq2 4966 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑦𝐵𝑥𝑦𝐵𝑧))
1211equsexvw 1988 . . . . . . . . . 10 (∃𝑥(𝑥 = 𝑧𝑦𝐵𝑥) ↔ 𝑦𝐵𝑧)
137, 10, 123bitri 298 . . . . . . . . 9 (𝑦( I ∘ 𝐵)𝑧𝑦𝐵𝑧)
1413a1i 11 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑦( I ∘ 𝐵)𝑧𝑦𝐵𝑧))
15 eleq1 2870 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ ( I ∘ 𝐵)))
16 df-br 4963 . . . . . . . . 9 (𝑦( I ∘ 𝐵)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ( I ∘ 𝐵))
1715, 16syl6bbr 290 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑦( I ∘ 𝐵)𝑧))
18 eleq1 2870 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥𝐵 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐵))
19 df-br 4963 . . . . . . . . 9 (𝑦𝐵𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐵)
2018, 19syl6bbr 290 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥𝐵𝑦𝐵𝑧))
2114, 17, 203bitr4d 312 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑥𝐵))
2221exlimivv 1910 . . . . . 6 (∃𝑦𝑧 𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑥𝐵))
234, 22syl 17 . . . . 5 ((Rel 𝐴𝑥𝐴) → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑥𝐵))
2423pm5.74da 800 . . . 4 (Rel 𝐴 → ((𝑥𝐴𝑥 ∈ ( I ∘ 𝐵)) ↔ (𝑥𝐴𝑥𝐵)))
2524albidv 1898 . . 3 (Rel 𝐴 → (∀𝑥(𝑥𝐴𝑥 ∈ ( I ∘ 𝐵)) ↔ ∀𝑥(𝑥𝐴𝑥𝐵)))
26 dfss2 3877 . . 3 (𝐴 ⊆ ( I ∘ 𝐵) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ( I ∘ 𝐵)))
27 dfss2 3877 . . 3 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
2825, 26, 273bitr4g 315 . 2 (Rel 𝐴 → (𝐴 ⊆ ( I ∘ 𝐵) ↔ 𝐴𝐵))
293, 28biadanii 819 1 (𝐴 ⊆ ( I ∘ 𝐵) ↔ (Rel 𝐴𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1520   = wceq 1522  ∃wex 1761   ∈ wcel 2081   ⊆ wss 3859  ⟨cop 4478   class class class wbr 4962   I cid 5347   ∘ ccom 5447  Rel wrel 5448 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-id 5348  df-xp 5449  df-rel 5450  df-co 5452 This theorem is referenced by:  dffun10  32984
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