Theorem ssrelrel 4440

Description: A subclass relationship determined by ordered triples. Use relrelss 4844 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 2939	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵))
2	1	alrimiv 1754	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵))
3	2	alrimivv 1755	. 2 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵))
4		elvvv 4403	. . . . . . . 8 ⊢ (𝑤 ∈ ((V × V) × V) ↔ ∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
5		eleq1 2100	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴))
6		eleq1 2100	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐵 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵))
7	5, 6	imbi12d 223	. . . . . . . . . . . . 13 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → ((𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵) ↔ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵)))
8	7	biimprcd 149	. . . . . . . . . . . 12 ⊢ ((⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)))
9	8	alimi 1344	. . . . . . . . . . 11 ⊢ (∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → ∀𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)))
10		19.23v 1763	. . . . . . . . . . 11 ⊢ (∀𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)) ↔ (∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)))
11	9, 10	sylib 127	. . . . . . . . . 10 ⊢ (∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)))
12	11	2alimi 1345	. . . . . . . . 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 1764	. . . . . . . . 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 127	. . . . . . . 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 141	. . . . . . 7 ⊢ (∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (𝑤 ∈ ((V × V) × V) → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵)))
16	15	com23 72	. . . . . 6 ⊢ (∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (𝑤 ∈ 𝐴 → (𝑤 ∈ ((V × V) × V) → 𝑤 ∈ 𝐵)))
17	16	a2d 23	. . . . 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 1759	. . . 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 2934	. . . 4 ⊢ (𝐴 ⊆ ((V × V) × V) ↔ ∀𝑤(𝑤 ∈ 𝐴 → 𝑤 ∈ ((V × V) × V)))
20		dfss2 2934	. . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑤(𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵))
21	18, 19, 20	3imtr4g 194	. . 3 ⊢ (∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (𝐴 ⊆ ((V × V) × V) → 𝐴 ⊆ 𝐵))
22	21	com12 27	. 2 ⊢ (𝐴 ⊆ ((V × V) × V) → (∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → 𝐴 ⊆ 𝐵))
23	3, 22	impbid2 131	1 ⊢ (𝐴 ⊆ ((V × V) × V) → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241   = wceq 1243  ∃wex 1381   ∈ wcel 1393  Vcvv 2557   ⊆ wss 2917  ⟨cop 3378   × cxp 4343

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-xp 4351

This theorem is referenced by:  eqrelrel  4441