Theorem ssrel 4845

Description: A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by NM, 2-Aug-1994.) (Revised by set.mm contributors, 27-Aug-2011.)

Assertion

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

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

Proof of Theorem ssrel

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3268	. . 3 ⊢ (A ⊆ B → (⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B))
2	1	alrimivv 1632	. 2 ⊢ (A ⊆ B → ∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B))
3		vex 2863	. . . . . . 7 ⊢ z ∈ V
4	3	proj1ex 4594	. . . . . 6 ⊢ Proj1 z ∈ V
5		opeq1 4579	. . . . . . . . 9 ⊢ (x = Proj1 z → ⟨x, y⟩ = ⟨ Proj1 z, y⟩)
6	5	eleq1d 2419	. . . . . . . 8 ⊢ (x = Proj1 z → (⟨x, y⟩ ∈ A ↔ ⟨ Proj1 z, y⟩ ∈ A))
7	5	eleq1d 2419	. . . . . . . 8 ⊢ (x = Proj1 z → (⟨x, y⟩ ∈ B ↔ ⟨ Proj1 z, y⟩ ∈ B))
8	6, 7	imbi12d 311	. . . . . . 7 ⊢ (x = Proj1 z → ((⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) ↔ (⟨ Proj1 z, y⟩ ∈ A → ⟨ Proj1 z, y⟩ ∈ B)))
9	8	albidv 1625	. . . . . 6 ⊢ (x = Proj1 z → (∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) ↔ ∀y(⟨ Proj1 z, y⟩ ∈ A → ⟨ Proj1 z, y⟩ ∈ B)))
10	4, 9	spcv 2946	. . . . 5 ⊢ (∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → ∀y(⟨ Proj1 z, y⟩ ∈ A → ⟨ Proj1 z, y⟩ ∈ B))
11	3	proj2ex 4595	. . . . . 6 ⊢ Proj2 z ∈ V
12		opeq2 4580	. . . . . . . 8 ⊢ (y = Proj2 z → ⟨ Proj1 z, y⟩ = ⟨ Proj1 z, Proj2 z⟩)
13	12	eleq1d 2419	. . . . . . 7 ⊢ (y = Proj2 z → (⟨ Proj1 z, y⟩ ∈ A ↔ ⟨ Proj1 z, Proj2 z⟩ ∈ A))
14	12	eleq1d 2419	. . . . . . 7 ⊢ (y = Proj2 z → (⟨ Proj1 z, y⟩ ∈ B ↔ ⟨ Proj1 z, Proj2 z⟩ ∈ B))
15	13, 14	imbi12d 311	. . . . . 6 ⊢ (y = Proj2 z → ((⟨ Proj1 z, y⟩ ∈ A → ⟨ Proj1 z, y⟩ ∈ B) ↔ (⟨ Proj1 z, Proj2 z⟩ ∈ A → ⟨ Proj1 z, Proj2 z⟩ ∈ B)))
16	11, 15	spcv 2946	. . . . 5 ⊢ (∀y(⟨ Proj1 z, y⟩ ∈ A → ⟨ Proj1 z, y⟩ ∈ B) → (⟨ Proj1 z, Proj2 z⟩ ∈ A → ⟨ Proj1 z, Proj2 z⟩ ∈ B))
17	10, 16	syl 15	. . . 4 ⊢ (∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → (⟨ Proj1 z, Proj2 z⟩ ∈ A → ⟨ Proj1 z, Proj2 z⟩ ∈ B))
18		opeq 4620	. . . . 5 ⊢ z = ⟨ Proj1 z, Proj2 z⟩
19	18	eleq1i 2416	. . . 4 ⊢ (z ∈ A ↔ ⟨ Proj1 z, Proj2 z⟩ ∈ A)
20	18	eleq1i 2416	. . . 4 ⊢ (z ∈ B ↔ ⟨ Proj1 z, Proj2 z⟩ ∈ B)
21	17, 19, 20	3imtr4g 261	. . 3 ⊢ (∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → (z ∈ A → z ∈ B))
22	21	ssrdv 3279	. 2 ⊢ (∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) → A ⊆ B)
23	2, 22	impbii 180	1 ⊢ (A ⊆ B ↔ ∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710   ⊆ wss 3258  ⟨cop 4562   Proj1 cproj1 4564   Proj2 cproj2 4565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-0c 4378  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569

This theorem is referenced by:  eqrel  4846  ssopr  4847  relssi  4849  relssdv  4850  cotr  5027  cnvsym  5028  intasym  5029  intirr  5030  ssdmrn  5100  dffun2  5120  fvfullfunlem2  5863