Theorem ssopr 4847

Description: Subclass principle for operators. (Contributed by Scott Fenton, 19-Apr-2021.)

Assertion

Ref	Expression
ssopr	⊢ (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 ssopr

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrel 4845	. . 3 ⊢ (A ⊆ B ↔ ∀w∀z(⟨w, z⟩ ∈ A → ⟨w, z⟩ ∈ B))
2		alcom 1737	. . 3 ⊢ (∀w∀z(⟨w, z⟩ ∈ A → ⟨w, z⟩ ∈ B) ↔ ∀z∀w(⟨w, z⟩ ∈ A → ⟨w, z⟩ ∈ B))
3	1, 2	bitri 240	. 2 ⊢ (A ⊆ B ↔ ∀z∀w(⟨w, z⟩ ∈ A → ⟨w, z⟩ ∈ B))
4		vex 2863	. . . . . . . 8 ⊢ w ∈ V
5		opeqex 4622	. . . . . . . 8 ⊢ (w ∈ V → ∃x∃y w = ⟨x, y⟩)
6	4, 5	ax-mp 5	. . . . . . 7 ⊢ ∃x∃y w = ⟨x, y⟩
7	6	a1bi 327	. . . . . 6 ⊢ ((⟨w, z⟩ ∈ A → ⟨w, z⟩ ∈ B) ↔ (∃x∃y w = ⟨x, y⟩ → (⟨w, z⟩ ∈ A → ⟨w, z⟩ ∈ B)))
8		19.23vv 1892	. . . . . 6 ⊢ (∀x∀y(w = ⟨x, y⟩ → (⟨w, z⟩ ∈ A → ⟨w, z⟩ ∈ B)) ↔ (∃x∃y w = ⟨x, y⟩ → (⟨w, z⟩ ∈ A → ⟨w, z⟩ ∈ B)))
9	7, 8	bitr4i 243	. . . . 5 ⊢ ((⟨w, z⟩ ∈ A → ⟨w, z⟩ ∈ B) ↔ ∀x∀y(w = ⟨x, y⟩ → (⟨w, z⟩ ∈ A → ⟨w, z⟩ ∈ B)))
10	9	albii 1566	. . . 4 ⊢ (∀w(⟨w, z⟩ ∈ A → ⟨w, z⟩ ∈ B) ↔ ∀w∀x∀y(w = ⟨x, y⟩ → (⟨w, z⟩ ∈ A → ⟨w, z⟩ ∈ B)))
11		alrot3 1738	. . . 4 ⊢ (∀w∀x∀y(w = ⟨x, y⟩ → (⟨w, z⟩ ∈ A → ⟨w, z⟩ ∈ B)) ↔ ∀x∀y∀w(w = ⟨x, y⟩ → (⟨w, z⟩ ∈ A → ⟨w, z⟩ ∈ B)))
12		vex 2863	. . . . . . 7 ⊢ x ∈ V
13		vex 2863	. . . . . . 7 ⊢ y ∈ V
14	12, 13	opex 4589	. . . . . 6 ⊢ ⟨x, y⟩ ∈ V
15		opeq1 4579	. . . . . . . 8 ⊢ (w = ⟨x, y⟩ → ⟨w, z⟩ = ⟨⟨x, y⟩, z⟩)
16	15	eleq1d 2419	. . . . . . 7 ⊢ (w = ⟨x, y⟩ → (⟨w, z⟩ ∈ A ↔ ⟨⟨x, y⟩, z⟩ ∈ A))
17	15	eleq1d 2419	. . . . . . 7 ⊢ (w = ⟨x, y⟩ → (⟨w, z⟩ ∈ B ↔ ⟨⟨x, y⟩, z⟩ ∈ B))
18	16, 17	imbi12d 311	. . . . . 6 ⊢ (w = ⟨x, y⟩ → ((⟨w, z⟩ ∈ A → ⟨w, z⟩ ∈ B) ↔ (⟨⟨x, y⟩, z⟩ ∈ A → ⟨⟨x, y⟩, z⟩ ∈ B)))
19	14, 18	ceqsalv 2886	. . . . 5 ⊢ (∀w(w = ⟨x, y⟩ → (⟨w, z⟩ ∈ A → ⟨w, z⟩ ∈ B)) ↔ (⟨⟨x, y⟩, z⟩ ∈ A → ⟨⟨x, y⟩, z⟩ ∈ B))
20	19	2albii 1567	. . . 4 ⊢ (∀x∀y∀w(w = ⟨x, y⟩ → (⟨w, z⟩ ∈ A → ⟨w, z⟩ ∈ B)) ↔ ∀x∀y(⟨⟨x, y⟩, z⟩ ∈ A → ⟨⟨x, y⟩, z⟩ ∈ B))
21	10, 11, 20	3bitri 262	. . 3 ⊢ (∀w(⟨w, z⟩ ∈ A → ⟨w, z⟩ ∈ B) ↔ ∀x∀y(⟨⟨x, y⟩, z⟩ ∈ A → ⟨⟨x, y⟩, z⟩ ∈ B))
22	21	albii 1566	. 2 ⊢ (∀z∀w(⟨w, z⟩ ∈ A → ⟨w, z⟩ ∈ B) ↔ ∀z∀x∀y(⟨⟨x, y⟩, z⟩ ∈ A → ⟨⟨x, y⟩, z⟩ ∈ B))
23		alrot3 1738	. 2 ⊢ (∀z∀x∀y(⟨⟨x, y⟩, z⟩ ∈ A → ⟨⟨x, y⟩, z⟩ ∈ B) ↔ ∀x∀y∀z(⟨⟨x, y⟩, z⟩ ∈ A → ⟨⟨x, y⟩, z⟩ ∈ B))
24	3, 22, 23	3bitri 262	1 ⊢ (A ⊆ B ↔ ∀x∀y∀z(⟨⟨x, y⟩, z⟩ ∈ A → ⟨⟨x, y⟩, z⟩ ∈ B))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ⊆ wss 3258  ⟨cop 4562

This proof depends on 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-13 1712  ax-14 1714  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 proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  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-pss 3262  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-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569

This theorem is used by:  eqopr  4848