Theorem sscoid 33374

Description: A condition for subset and composition with identity. (Contributed by Scott Fenton, 13-Apr-2018.)

Assertion

Ref	Expression
sscoid	⊢ (𝐴 ⊆ ( I ∘ 𝐵) ↔ (Rel 𝐴 ∧ 𝐴 ⊆ 𝐵))

Proof of Theorem sscoid

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relco 6097	. . 3 ⊢ Rel ( I ∘ 𝐵)
2		relss 5656	. . 3 ⊢ (𝐴 ⊆ ( I ∘ 𝐵) → (Rel ( I ∘ 𝐵) → Rel 𝐴))
3	1, 2	mpi 20	. 2 ⊢ (𝐴 ⊆ ( I ∘ 𝐵) → Rel 𝐴)
4		elrel 5671	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑦∃𝑧 𝑥 = ⟨𝑦, 𝑧⟩)
5		vex 3497	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
6		vex 3497	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
7	5, 6	brco 5741	. . . . . . . . . 10 ⊢ (𝑦( I ∘ 𝐵)𝑧 ↔ ∃𝑥(𝑦𝐵𝑥 ∧ 𝑥 I 𝑧))
8	6	ideq 5723	. . . . . . . . . . . 12 ⊢ (𝑥 I 𝑧 ↔ 𝑥 = 𝑧)
9	8	anbi1ci 627	. . . . . . . . . . 11 ⊢ ((𝑦𝐵𝑥 ∧ 𝑥 I 𝑧) ↔ (𝑥 = 𝑧 ∧ 𝑦𝐵𝑥))
10	9	exbii 1848	. . . . . . . . . 10 ⊢ (∃𝑥(𝑦𝐵𝑥 ∧ 𝑥 I 𝑧) ↔ ∃𝑥(𝑥 = 𝑧 ∧ 𝑦𝐵𝑥))
11		breq2 5070	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑦𝐵𝑥 ↔ 𝑦𝐵𝑧))
12	11	equsexvw 2011	. . . . . . . . . 10 ⊢ (∃𝑥(𝑥 = 𝑧 ∧ 𝑦𝐵𝑥) ↔ 𝑦𝐵𝑧)
13	7, 10, 12	3bitri 299	. . . . . . . . 9 ⊢ (𝑦( I ∘ 𝐵)𝑧 ↔ 𝑦𝐵𝑧)
14	13	a1i 11	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑦( I ∘ 𝐵)𝑧 ↔ 𝑦𝐵𝑧))
15		eleq1 2900	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ ( I ∘ 𝐵)))
16		df-br 5067	. . . . . . . . 9 ⊢ (𝑦( I ∘ 𝐵)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ( I ∘ 𝐵))
17	15, 16	syl6bbr 291	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑦( I ∘ 𝐵)𝑧))
18		eleq1 2900	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ 𝐵 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐵))
19		df-br 5067	. . . . . . . . 9 ⊢ (𝑦𝐵𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐵)
20	18, 19	syl6bbr 291	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ 𝐵 ↔ 𝑦𝐵𝑧))
21	14, 17, 20	3bitr4d 313	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑥 ∈ 𝐵))
22	21	exlimivv 1933	. . . . . 6 ⊢ (∃𝑦∃𝑧 𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑥 ∈ 𝐵))
23	4, 22	syl 17	. . . . 5 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑥 ∈ 𝐵))
24	23	pm5.74da 802	. . . 4 ⊢ (Rel 𝐴 → ((𝑥 ∈ 𝐴 → 𝑥 ∈ ( I ∘ 𝐵)) ↔ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)))
25	24	albidv 1921	. . 3 ⊢ (Rel 𝐴 → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ( I ∘ 𝐵)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)))
26		dfss2 3955	. . 3 ⊢ (𝐴 ⊆ ( I ∘ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ( I ∘ 𝐵)))
27		dfss2 3955	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
28	25, 26, 27	3bitr4g 316	. 2 ⊢ (Rel 𝐴 → (𝐴 ⊆ ( I ∘ 𝐵) ↔ 𝐴 ⊆ 𝐵))
29	3, 28	biadanii 820	1 ⊢ (𝐴 ⊆ ( I ∘ 𝐵) ↔ (Rel 𝐴 ∧ 𝐴 ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ⊆ wss 3936  ⟨cop 4573   class class class wbr 5066   I cid 5459   ∘ ccom 5559  Rel wrel 5560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-co 5564

This theorem is referenced by:  dffun10  33375