Theorem sscoid 36369

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 6110	. . 3 ⊢ Rel ( I ∘ 𝐵)
2		relss 5768	. . 3 ⊢ (𝐴 ⊆ ( I ∘ 𝐵) → (Rel ( I ∘ 𝐵) → Rel 𝐴))
3	1, 2	mpi 21	. 2 ⊢ (𝐴 ⊆ ( I ∘ 𝐵) → Rel 𝐴)
4		elrel 5784	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑦∃𝑧 𝑥 = ⟨𝑦, 𝑧⟩)
5		vex 3457	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
6		vex 3457	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
7	5, 6	brco 5856	. . . . . . . . . 10 ⊢ (𝑦( I ∘ 𝐵)𝑧 ↔ ∃𝑥(𝑦𝐵𝑥 ∧ 𝑥 I 𝑧))
8	6	ideq 5838	. . . . . . . . . . . 12 ⊢ (𝑥 I 𝑧 ↔ 𝑥 = 𝑧)
9	8	anbi1ci 637	. . . . . . . . . . 11 ⊢ ((𝑦𝐵𝑥 ∧ 𝑥 I 𝑧) ↔ (𝑥 = 𝑧 ∧ 𝑦𝐵𝑥))
10	9	exbii 1876	. . . . . . . . . 10 ⊢ (∃𝑥(𝑦𝐵𝑥 ∧ 𝑥 I 𝑧) ↔ ∃𝑥(𝑥 = 𝑧 ∧ 𝑦𝐵𝑥))
11		breq2 5112	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑦𝐵𝑥 ↔ 𝑦𝐵𝑧))
12	11	equsexvw 2033	. . . . . . . . . 10 ⊢ (∃𝑥(𝑥 = 𝑧 ∧ 𝑦𝐵𝑥) ↔ 𝑦𝐵𝑧)
13	7, 10, 12	3bitri 300	. . . . . . . . 9 ⊢ (𝑦( I ∘ 𝐵)𝑧 ↔ 𝑦𝐵𝑧)
14	13	a1i 11	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑦( I ∘ 𝐵)𝑧 ↔ 𝑦𝐵𝑧))
15		eleq1 2849	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ ( I ∘ 𝐵)))
16		df-br 5109	. . . . . . . . 9 ⊢ (𝑦( I ∘ 𝐵)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ( I ∘ 𝐵))
17	15, 16	bitr4di 292	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑦( I ∘ 𝐵)𝑧))
18		eleq1 2849	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ 𝐵 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐵))
19		df-br 5109	. . . . . . . . 9 ⊢ (𝑦𝐵𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐵)
20	18, 19	bitr4di 292	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ 𝐵 ↔ 𝑦𝐵𝑧))
21	14, 17, 20	3bitr4d 314	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑥 ∈ 𝐵))
22	21	exlimivv 1960	. . . . . 6 ⊢ (∃𝑦∃𝑧 𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑥 ∈ 𝐵))
23	4, 22	syl 18	. . . . 5 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑥 ∈ 𝐵))
24	23	pm5.74da 815	. . . 4 ⊢ (Rel 𝐴 → ((𝑥 ∈ 𝐴 → 𝑥 ∈ ( I ∘ 𝐵)) ↔ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)))
25	24	albidv 1948	. . 3 ⊢ (Rel 𝐴 → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ( I ∘ 𝐵)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)))
26		df-ss 3921	. . 3 ⊢ (𝐴 ⊆ ( I ∘ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ( I ∘ 𝐵)))
27		df-ss 3921	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
28	25, 26, 27	3bitr4g 317	. 2 ⊢ (Rel 𝐴 → (𝐴 ⊆ ( I ∘ 𝐵) ↔ 𝐴 ⊆ 𝐵))
29	3, 28	biadanii 833	1 ⊢ (𝐴 ⊆ ( I ∘ 𝐵) ↔ (Rel 𝐴 ∧ 𝐴 ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1566   = wceq 1568  ∃wex 1807   ∈ wcel 2141   ⊆ wss 3904  ⟨cop 4594   class class class wbr 5108   I cid 5555   ∘ ccom 5665  Rel wrel 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-id 5556  df-xp 5667  df-rel 5668  df-co 5670

This theorem is referenced by:  dffun10  36370