Theorem sscoid 35901

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 6079	. . 3 ⊢ Rel ( I ∘ 𝐵)
2		relss 5744	. . 3 ⊢ (𝐴 ⊆ ( I ∘ 𝐵) → (Rel ( I ∘ 𝐵) → Rel 𝐴))
3	1, 2	mpi 20	. 2 ⊢ (𝐴 ⊆ ( I ∘ 𝐵) → Rel 𝐴)
4		elrel 5761	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑦∃𝑧 𝑥 = ⟨𝑦, 𝑧⟩)
5		vex 3451	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
6		vex 3451	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
7	5, 6	brco 5834	. . . . . . . . . 10 ⊢ (𝑦( I ∘ 𝐵)𝑧 ↔ ∃𝑥(𝑦𝐵𝑥 ∧ 𝑥 I 𝑧))
8	6	ideq 5816	. . . . . . . . . . . 12 ⊢ (𝑥 I 𝑧 ↔ 𝑥 = 𝑧)
9	8	anbi1ci 626	. . . . . . . . . . 11 ⊢ ((𝑦𝐵𝑥 ∧ 𝑥 I 𝑧) ↔ (𝑥 = 𝑧 ∧ 𝑦𝐵𝑥))
10	9	exbii 1848	. . . . . . . . . 10 ⊢ (∃𝑥(𝑦𝐵𝑥 ∧ 𝑥 I 𝑧) ↔ ∃𝑥(𝑥 = 𝑧 ∧ 𝑦𝐵𝑥))
11		breq2 5111	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑦𝐵𝑥 ↔ 𝑦𝐵𝑧))
12	11	equsexvw 2005	. . . . . . . . . 10 ⊢ (∃𝑥(𝑥 = 𝑧 ∧ 𝑦𝐵𝑥) ↔ 𝑦𝐵𝑧)
13	7, 10, 12	3bitri 297	. . . . . . . . 9 ⊢ (𝑦( I ∘ 𝐵)𝑧 ↔ 𝑦𝐵𝑧)
14	13	a1i 11	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑦( I ∘ 𝐵)𝑧 ↔ 𝑦𝐵𝑧))
15		eleq1 2816	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ ( I ∘ 𝐵)))
16		df-br 5108	. . . . . . . . 9 ⊢ (𝑦( I ∘ 𝐵)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ( I ∘ 𝐵))
17	15, 16	bitr4di 289	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑦( I ∘ 𝐵)𝑧))
18		eleq1 2816	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ 𝐵 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐵))
19		df-br 5108	. . . . . . . . 9 ⊢ (𝑦𝐵𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐵)
20	18, 19	bitr4di 289	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ 𝐵 ↔ 𝑦𝐵𝑧))
21	14, 17, 20	3bitr4d 311	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑥 ∈ 𝐵))
22	21	exlimivv 1932	. . . . . 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 803	. . . 4 ⊢ (Rel 𝐴 → ((𝑥 ∈ 𝐴 → 𝑥 ∈ ( I ∘ 𝐵)) ↔ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)))
25	24	albidv 1920	. . 3 ⊢ (Rel 𝐴 → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ( I ∘ 𝐵)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)))
26		df-ss 3931	. . 3 ⊢ (𝐴 ⊆ ( I ∘ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ( I ∘ 𝐵)))
27		df-ss 3931	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
28	25, 26, 27	3bitr4g 314	. 2 ⊢ (Rel 𝐴 → (𝐴 ⊆ ( I ∘ 𝐵) ↔ 𝐴 ⊆ 𝐵))
29	3, 28	biadanii 821	1 ⊢ (𝐴 ⊆ ( I ∘ 𝐵) ↔ (Rel 𝐴 ∧ 𝐴 ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ⊆ wss 3914  ⟨cop 4595   class class class wbr 5107   I cid 5532   ∘ ccom 5642  Rel wrel 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-co 5647

This theorem is referenced by:  dffun10  35902