Theorem ssrelrel 5669

Description: A subclass relationship determined by ordered triples. Use relrelss 6124 to express the antecedent in terms of the relation predicate. (Contributed by NM, 17-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
ssrelrel	⊢ (𝐴 ⊆ ((V × V) × V) → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧

Proof of Theorem ssrelrel

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3961	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵))
2	1	alrimiv 1928	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵))
3	2	alrimivv 1929	. 2 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵))
4		elvvv 5627	. . . . . . . 8 ⊢ (𝑤 ∈ ((V × V) × V) ↔ ∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
5		eleq1 2900	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴))
6		eleq1 2900	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐵 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵))
7	5, 6	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → ((𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵) ↔ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵)))
8	7	biimprcd 252	. . . . . . . . . . . 12 ⊢ ((⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)))
9	8	alimi 1812	. . . . . . . . . . 11 ⊢ (∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → ∀𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)))
10		19.23v 1943	. . . . . . . . . . 11 ⊢ (∀𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)) ↔ (∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)))
11	9, 10	sylib 220	. . . . . . . . . 10 ⊢ (∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)))
12	11	2alimi 1813	. . . . . . . . 9 ⊢ (∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → ∀𝑥∀𝑦(∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)))
13		19.23vv 1944	. . . . . . . . 9 ⊢ (∀𝑥∀𝑦(∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)) ↔ (∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)))
14	12, 13	sylib 220	. . . . . . . 8 ⊢ (∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)))
15	4, 14	syl5bi 244	. . . . . . 7 ⊢ (∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (𝑤 ∈ ((V × V) × V) → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)))
16	15	com23 86	. . . . . 6 ⊢ (∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (𝑤 ∈ 𝐴 → (𝑤 ∈ ((V × V) × V) → 𝑤 ∈ 𝐵)))
17	16	a2d 29	. . . . 5 ⊢ (∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → ((𝑤 ∈ 𝐴 → 𝑤 ∈ ((V × V) × V)) → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)))
18	17	alimdv 1917	. . . 4 ⊢ (∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (∀𝑤(𝑤 ∈ 𝐴 → 𝑤 ∈ ((V × V) × V)) → ∀𝑤(𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)))
19		dfss2 3955	. . . 4 ⊢ (𝐴 ⊆ ((V × V) × V) ↔ ∀𝑤(𝑤 ∈ 𝐴 → 𝑤 ∈ ((V × V) × V)))
20		dfss2 3955	. . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑤(𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵))
21	18, 19, 20	3imtr4g 298	. . 3 ⊢ (∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (𝐴 ⊆ ((V × V) × V) → 𝐴 ⊆ 𝐵))
22	21	com12 32	. 2 ⊢ (𝐴 ⊆ ((V × V) × V) → (∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → 𝐴 ⊆ 𝐵))
23	3, 22	impbid2 228	1 ⊢ (𝐴 ⊆ ((V × V) × V) → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114  Vcvv 3494   ⊆ wss 3936  ⟨cop 4573   × cxp 5553

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-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  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-opab 5129  df-xp 5561

This theorem is referenced by:  eqrelrel  5670