Theorem ssrelrel 4363

Description: A subclass relationship determined by ordered triples. Use relrelss 4767 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 2912	. . . 4 ⊢ (A ⊆ B → (⟨⟨x, y⟩, z⟩ ∈ A → ⟨⟨x, y⟩, z⟩ ∈ B))
2	1	alrimiv 1732	. . 3 ⊢ (A ⊆ B → ∀z(⟨⟨x, y⟩, z⟩ ∈ A → ⟨⟨x, y⟩, z⟩ ∈ B))
3	2	alrimivv 1733	. 2 ⊢ (A ⊆ B → ∀x∀y∀z(⟨⟨x, y⟩, z⟩ ∈ A → ⟨⟨x, y⟩, z⟩ ∈ B))
4		elvvv 4326	. . . . . . . 8 ⊢ (w ∈ ((V × V) × V) ↔ ∃x∃y∃z w = ⟨⟨x, y⟩, z⟩)
5		eleq1 2078	. . . . . . . . . . . . . 14 ⊢ (w = ⟨⟨x, y⟩, z⟩ → (w ∈ A ↔ ⟨⟨x, y⟩, z⟩ ∈ A))
6		eleq1 2078	. . . . . . . . . . . . . 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 1320	. . . . . . . . . . 11 ⊢ (∀z(⟨⟨x, y⟩, z⟩ ∈ A → ⟨⟨x, y⟩, z⟩ ∈ B) → ∀z(w = ⟨⟨x, y⟩, z⟩ → (w ∈ A → w ∈ B)))
10		19.23v 1741	. . . . . . . . . . 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 1321	. . . . . . . . 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 1742	. . . . . . . . 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 1737	. . . 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 2907	. . . 4 ⊢ (A ⊆ ((V × V) × V) ↔ ∀w(w ∈ A → w ∈ ((V × V) × V)))
20		dfss2 2907	. . . 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 1224   = wceq 1226  ∃wex 1358   ∈ wcel 1370  Vcvv 2531   ⊆ wss 2890  ⟨cop 3349   × cxp 4266

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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914

This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-opab 3789  df-xp 4274

This theorem is referenced by:  eqrelrel  4364