Theorem ssrelrel 5797

Description: A subclass relationship determined by ordered triples. Use relrelss 6273 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 3976	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵))
2	1	alrimiv 1931	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵))
3	2	alrimivv 1932	. 2 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵))
4		elvvv 5752	. . . . . . . 8 ⊢ (𝑤 ∈ ((V × V) × V) ↔ ∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
5		eleq1 2822	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴))
6		eleq1 2822	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐵 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵))
7	5, 6	imbi12d 345	. . . . . . . . . . . . 13 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → ((𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵) ↔ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵)))
8	7	biimprcd 249	. . . . . . . . . . . 12 ⊢ ((⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)))
9	8	alimi 1814	. . . . . . . . . . 11 ⊢ (∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → ∀𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)))
10		19.23v 1946	. . . . . . . . . . 11 ⊢ (∀𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)) ↔ (∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)))
11	9, 10	sylib 217	. . . . . . . . . 10 ⊢ (∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)))
12	11	2alimi 1815	. . . . . . . . 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 1947	. . . . . . . . 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 217	. . . . . . . 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	biimtrid 241	. . . . . . 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 1920	. . . 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 3969	. . . 4 ⊢ (𝐴 ⊆ ((V × V) × V) ↔ ∀𝑤(𝑤 ∈ 𝐴 → 𝑤 ∈ ((V × V) × V)))
20		dfss2 3969	. . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑤(𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵))
21	18, 19, 20	3imtr4g 296	. . 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 225	1 ⊢ (𝐴 ⊆ ((V × V) × V) → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1540   = wceq 1542  ∃wex 1782   ∈ wcel 2107  Vcvv 3475   ⊆ wss 3949  ⟨cop 4635   × cxp 5675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5212  df-xp 5683

This theorem is referenced by:  eqrelrel  5798