Theorem ssrelrel 4763

Description: A subclass relationship determined by ordered triples. Use relrelss 5196 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 3177	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵))
2	1	alrimiv 1888	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵))
3	2	alrimivv 1889	. 2 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵))
4		elvvv 4726	. . . . . . . 8 ⊢ (𝑤 ∈ ((V × V) × V) ↔ ∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
5		eleq1 2259	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴))
6		eleq1 2259	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐵 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵))
7	5, 6	imbi12d 234	. . . . . . . . . . . . 13 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → ((𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵) ↔ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵)))
8	7	biimprcd 160	. . . . . . . . . . . 12 ⊢ ((⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)))
9	8	alimi 1469	. . . . . . . . . . 11 ⊢ (∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → ∀𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)))
10		19.23v 1897	. . . . . . . . . . 11 ⊢ (∀𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)) ↔ (∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)))
11	9, 10	sylib 122	. . . . . . . . . 10 ⊢ (∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)))
12	11	2alimi 1470	. . . . . . . . 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 1898	. . . . . . . . 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 122	. . . . . . . 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 152	. . . . . . 7 ⊢ (∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (𝑤 ∈ ((V × V) × V) → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)))
16	15	com23 78	. . . . . 6 ⊢ (∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (𝑤 ∈ 𝐴 → (𝑤 ∈ ((V × V) × V) → 𝑤 ∈ 𝐵)))
17	16	a2d 26	. . . . 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 1893	. . . 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 3172	. . . 4 ⊢ (𝐴 ⊆ ((V × V) × V) ↔ ∀𝑤(𝑤 ∈ 𝐴 → 𝑤 ∈ ((V × V) × V)))
20		dfss2 3172	. . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑤(𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵))
21	18, 19, 20	3imtr4g 205	. . 3 ⊢ (∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (𝐴 ⊆ ((V × V) × V) → 𝐴 ⊆ 𝐵))
22	21	com12 30	. 2 ⊢ (𝐴 ⊆ ((V × V) × V) → (∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → 𝐴 ⊆ 𝐵))
23	3, 22	impbid2 143	1 ⊢ (𝐴 ⊆ ((V × V) × V) → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362   = wceq 1364  ∃wex 1506   ∈ wcel 2167  Vcvv 2763   ⊆ wss 3157  ⟨cop 3625   × cxp 4661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-opab 4095  df-xp 4669

This theorem is referenced by:  eqrelrel  4764