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 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 5972	. . 3 ⊢ Rel ( I ∘ 𝐵)
2		relss 5542	. . 3 ⊢ (𝐴 ⊆ ( I ∘ 𝐵) → (Rel ( I ∘ 𝐵) → Rel 𝐴))
3	1, 2	mpi 20	. 2 ⊢ (𝐴 ⊆ ( I ∘ 𝐵) → Rel 𝐴)
4		elrel 5557	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑦∃𝑧 𝑥 = ⟨𝑦, 𝑧⟩)
5		vex 3440	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
6		vex 3440	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
7	5, 6	brco 5627	. . . . . . . . . 10 ⊢ (𝑦( I ∘ 𝐵)𝑧 ↔ ∃𝑥(𝑦𝐵𝑥 ∧ 𝑥 I 𝑧))
8	6	ideq 5609	. . . . . . . . . . . 12 ⊢ (𝑥 I 𝑧 ↔ 𝑥 = 𝑧)
9	8	anbi1ci 625	. . . . . . . . . . 11 ⊢ ((𝑦𝐵𝑥 ∧ 𝑥 I 𝑧) ↔ (𝑥 = 𝑧 ∧ 𝑦𝐵𝑥))
10	9	exbii 1829	. . . . . . . . . 10 ⊢ (∃𝑥(𝑦𝐵𝑥 ∧ 𝑥 I 𝑧) ↔ ∃𝑥(𝑥 = 𝑧 ∧ 𝑦𝐵𝑥))
11		breq2 4966	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑦𝐵𝑥 ↔ 𝑦𝐵𝑧))
12	11	equsexvw 1988	. . . . . . . . . 10 ⊢ (∃𝑥(𝑥 = 𝑧 ∧ 𝑦𝐵𝑥) ↔ 𝑦𝐵𝑧)
13	7, 10, 12	3bitri 298	. . . . . . . . 9 ⊢ (𝑦( I ∘ 𝐵)𝑧 ↔ 𝑦𝐵𝑧)
14	13	a1i 11	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑦( I ∘ 𝐵)𝑧 ↔ 𝑦𝐵𝑧))
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))
17	15, 16	syl6bbr 290	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑦( I ∘ 𝐵)𝑧))
18		eleq1 2870	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ 𝐵 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐵))
19		df-br 4963	. . . . . . . . 9 ⊢ (𝑦𝐵𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐵)
20	18, 19	syl6bbr 290	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ 𝐵 ↔ 𝑦𝐵𝑧))
21	14, 17, 20	3bitr4d 312	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ( I ∘ 𝐵) ↔ 𝑥 ∈ 𝐵))
22	21	exlimivv 1910	. . . . . 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 800	. . . 4 ⊢ (Rel 𝐴 → ((𝑥 ∈ 𝐴 → 𝑥 ∈ ( I ∘ 𝐵)) ↔ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)))
25	24	albidv 1898	. . 3 ⊢ (Rel 𝐴 → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ( I ∘ 𝐵)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)))
26		dfss2 3877	. . 3 ⊢ (𝐴 ⊆ ( I ∘ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ( I ∘ 𝐵)))
27		dfss2 3877	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
28	25, 26, 27	3bitr4g 315	. 2 ⊢ (Rel 𝐴 → (𝐴 ⊆ ( I ∘ 𝐵) ↔ 𝐴 ⊆ 𝐵))
29	3, 28	biadanii 819	1 ⊢ (𝐴 ⊆ ( I ∘ 𝐵) ↔ (Rel 𝐴 ∧ 𝐴 ⊆ 𝐵))

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