Theorem ssrel 3242

Description: A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33.

Assertion

Ref	Expression
ssrel	⊢ (Rel A → (A ⊆ B ↔ ∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B)))

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

Proof of Theorem ssrel

Step	Hyp	Ref	Expression
1		ssel 2059	. . . . 5 ⊢ (A ⊆ B → (⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B))
2	1	a1i 8	. . . 4 ⊢ (Rel A → (A ⊆ B → (⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B)))
3	2	19.21adv 1286	. . 3 ⊢ (Rel A → (A ⊆ B → ∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B)))
4	3	19.21adv 1286	. 2 ⊢ (Rel A → (A ⊆ B → ∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B)))
5		df-rel 3180	. . . . . . . 8 ⊢ (Rel A ↔ A ⊆ (V × V))
6		ssel 2059	. . . . . . . 8 ⊢ (A ⊆ (V × V) → (z ∈ A → z ∈ (V × V)))
7	5, 6	sylbi 199	. . . . . . 7 ⊢ (Rel A → (z ∈ A → z ∈ (V × V)))
8		elvv 3223	. . . . . . 7 ⊢ (z ∈ (V × V) ↔ ∃x∃y z = ⟨x, y⟩)
9	7, 8	syl6ib 212	. . . . . 6 ⊢ (Rel A → (z ∈ A → ∃x∃y z = ⟨x, y⟩))
10		id 59	. . . . . . . . . . . . . 14 ⊢ ((⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → (⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B))
11	10	anim2d 560	. . . . . . . . . . . . 13 ⊢ ((⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → ((z = ⟨x, y⟩ ⋀ ⟨x, y⟩ ∈ A) → (z = ⟨x, y⟩ ⋀ ⟨x, y⟩ ∈ B)))
12		eleq1 1531	. . . . . . . . . . . . . 14 ⊢ (z = ⟨x, y⟩ → (z ∈ B ↔ ⟨x, y⟩ ∈ B))
13	12	biimpar 417	. . . . . . . . . . . . 13 ⊢ ((z = ⟨x, y⟩ ⋀ ⟨x, y⟩ ∈ B) → z ∈ B)
14	11, 13	syl6 22	. . . . . . . . . . . 12 ⊢ ((⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → ((z = ⟨x, y⟩ ⋀ ⟨x, y⟩ ∈ A) → z ∈ B))
15		eleq1 1531	. . . . . . . . . . . . 13 ⊢ (z = ⟨x, y⟩ → (z ∈ A ↔ ⟨x, y⟩ ∈ A))
16	15	pm5.32i 644	. . . . . . . . . . . 12 ⊢ ((z = ⟨x, y⟩ ⋀ z ∈ A) ↔ (z = ⟨x, y⟩ ⋀ ⟨x, y⟩ ∈ A))
17	14, 16	syl5ib 206	. . . . . . . . . . 11 ⊢ ((⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → ((z = ⟨x, y⟩ ⋀ z ∈ A) → z ∈ B))
18	17	exp3a 375	. . . . . . . . . 10 ⊢ ((⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → (z = ⟨x, y⟩ → (z ∈ A → z ∈ B)))
19	18	19.20i 990	. . . . . . . . 9 ⊢ (∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → ∀y(z = ⟨x, y⟩ → (z ∈ A → z ∈ B)))
20		19.23v 1291	. . . . . . . . 9 ⊢ (∀y(z = ⟨x, y⟩ → (z ∈ A → z ∈ B)) ↔ (∃y z = ⟨x, y⟩ → (z ∈ A → z ∈ B)))
21	19, 20	sylib 198	. . . . . . . 8 ⊢ (∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → (∃y z = ⟨x, y⟩ → (z ∈ A → z ∈ B)))
22	21	19.20i 990	. . . . . . 7 ⊢ (∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → ∀x(∃y z = ⟨x, y⟩ → (z ∈ A → z ∈ B)))
23		19.23v 1291	. . . . . . 7 ⊢ (∀x(∃y z = ⟨x, y⟩ → (z ∈ A → z ∈ B)) ↔ (∃x∃y z = ⟨x, y⟩ → (z ∈ A → z ∈ B)))
24	22, 23	sylib 198	. . . . . 6 ⊢ (∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → (∃x∃y z = ⟨x, y⟩ → (z ∈ A → z ∈ B)))
25	9, 24	syl9 57	. . . . 5 ⊢ (Rel A → (∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → (z ∈ A → (z ∈ A → z ∈ B))))
26		pm2.43 63	. . . . 5 ⊢ ((z ∈ A → (z ∈ A → z ∈ B)) → (z ∈ A → z ∈ B))
27	25, 26	syl6 22	. . . 4 ⊢ (Rel A → (∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → (z ∈ A → z ∈ B)))
28	27	19.21adv 1286	. . 3 ⊢ (Rel A → (∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → ∀z(z ∈ A → z ∈ B)))
29		dfss2 2054	. . 3 ⊢ (A ⊆ B ↔ ∀z(z ∈ A → z ∈ B))
30	28, 29	syl6ibr 213	. 2 ⊢ (Rel A → (∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → A ⊆ B))
31	4, 30	impbid 515	1 ⊢ (Rel A → (A ⊆ B ↔ ∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B)))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 952   = wceq 954   ∈ wcel 956  ∃wex 978  Vcvv 1807   ⊆ wss 2043  ⟨cop 2407   × cxp 3163  Rel wrel 3170

This theorem is referenced by:  relssi 3243  relssdv 3244  eqrel 3245  intasym 3430

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-opab 2662  df-xp 3179  df-rel 3180

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