Theorem relss 2480

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

Assertion

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

Distinct variable group(s):   x,y,A   x,B,y

Proof of Theorem relss

Step	Hyp	Ref	Expression
1		ssel 1502	. . . . 5 ⊢ (A ⊆ B → (⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B))
2	1	a1i 7	. . . 4 ⊢ (Rel A → (A ⊆ B → (⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B)))
3	2	19.21adv 945	. . 3 ⊢ (Rel A → (A ⊆ B → ∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B)))
4	3	19.21adv 945	. 2 ⊢ (Rel A → (A ⊆ B → ∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B)))
5		df-rel 2425	. . . . . . . 8 ⊢ (Rel A ↔ A ⊆ (V × V))
6		ssel 1502	. . . . . . . 8 ⊢ (A ⊆ (V × V) → (z ∈ A → z ∈ (V × V)))
7	5, 6	sylbi 174	. . . . . . 7 ⊢ (Rel A → (z ∈ A → z ∈ (V × V)))
8		elvv 2464	. . . . . . 7 ⊢ (z ∈ (V × V) ↔ ∃x∃y z = ⟨x, y⟩)
9	7, 8	syl6ib 185	. . . . . 6 ⊢ (Rel A → (z ∈ A → ∃x∃y z = ⟨x, y⟩))
10		id 9	. . . . . . . . . . . . . 14 ⊢ ((⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → (⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B))
11	10	anim2d 433	. . . . . . . . . . . . 13 ⊢ ((⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → ((z = ⟨x, y⟩ ∧ ⟨x, y⟩ ∈ A) → (z = ⟨x, y⟩ ∧ ⟨x, y⟩ ∈ B)))
12		eleq1 1149	. . . . . . . . . . . . . 14 ⊢ (z = ⟨x, y⟩ → (z ∈ B ↔ ⟨x, y⟩ ∈ B))
13	12	biimpar 325	. . . . . . . . . . . . 13 ⊢ ((z = ⟨x, y⟩ ∧ ⟨x, y⟩ ∈ B) → z ∈ B)
14	11, 13	syl6 23	. . . . . . . . . . . 12 ⊢ ((⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → ((z = ⟨x, y⟩ ∧ ⟨x, y⟩ ∈ A) → z ∈ B))
15		eleq1 1149	. . . . . . . . . . . . 13 ⊢ (z = ⟨x, y⟩ → (z ∈ A ↔ ⟨x, y⟩ ∈ A))
16	15	pm5.32i 489	. . . . . . . . . . . 12 ⊢ ((z = ⟨x, y⟩ ∧ z ∈ A) ↔ (z = ⟨x, y⟩ ∧ ⟨x, y⟩ ∈ A))
17	14, 16	syl5ib 181	. . . . . . . . . . 11 ⊢ ((⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → ((z = ⟨x, y⟩ ∧ z ∈ A) → z ∈ B))
18	17	exp3a 292	. . . . . . . . . 10 ⊢ ((⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → (z = ⟨x, y⟩ → (z ∈ A → z ∈ B)))
19	18	19.20i 691	. . . . . . . . 9 ⊢ (∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → ∀y(z = ⟨x, y⟩ → (z ∈ A → z ∈ B)))
20		19.23v 950	. . . . . . . . 9 ⊢ (∀y(z = ⟨x, y⟩ → (z ∈ A → z ∈ B)) ↔ (∃y z = ⟨x, y⟩ → (z ∈ A → z ∈ B)))
21	19, 20	sylib 173	. . . . . . . 8 ⊢ (∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → (∃y z = ⟨x, y⟩ → (z ∈ A → z ∈ B)))
22	21	19.20i 691	. . . . . . 7 ⊢ (∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → ∀x(∃y z = ⟨x, y⟩ → (z ∈ A → z ∈ B)))
23		19.23v 950	. . . . . . 7 ⊢ (∀x(∃y z = ⟨x, y⟩ → (z ∈ A → z ∈ B)) ↔ (∃x∃y z = ⟨x, y⟩ → (z ∈ A → z ∈ B)))
24	22, 23	sylib 173	. . . . . 6 ⊢ (∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → (∃x∃y z = ⟨x, y⟩ → (z ∈ A → z ∈ B)))
25	9, 24	syl9 55	. . . . 5 ⊢ (Rel A → (∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → (z ∈ A → (z ∈ A → z ∈ B))))
26		pm2.43 57	. . . . 5 ⊢ ((z ∈ A → (z ∈ A → z ∈ B)) → (z ∈ A → z ∈ B))
27	25, 26	syl6 23	. . . 4 ⊢ (Rel A → (∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → (z ∈ A → z ∈ B)))
28	27	19.21adv 945	. . 3 ⊢ (Rel A → (∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → ∀z(z ∈ A → z ∈ B)))
29		dfss2 1497	. . 3 ⊢ (A ⊆ B ↔ ∀z(z ∈ A → z ∈ B))
30	28, 29	syl6ibr 186	. 2 ⊢ (Rel A → (∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → A ⊆ B))
31	4, 30	impbid 397	1 ⊢ (Rel A → (A ⊆ B ↔ ∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ⟨cop 1810   × cxp 2408  Rel wrel 2415

This theorem is referenced by:  relssi 2481  relssdv 2482  cleqrel 2483  intasym 2627

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424  df-rel 2425

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