setconslem4 - New Foundations Explorer

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Theorem setconslem4 4735

Description: Lemma for set construction functions. Create a mapping between the two types of ordered pair abstractions. (Contributed by SF, 7-Jan-2015.)

**Assertion**
Ref	Expression
setconslem4	⊢ ⋃₁⋃₁((((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c)) “_k A) = {⟨x, y⟩ ∣ ⟪x, y⟫ ∈ A}

Distinct variable group: x,A,y

Proof of Theorem setconslem4 Dummy variables z t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 4112	. . . . 5 ⊢ {{z}} ∈ V
2	1	elimak 4260	. . . 4 ⊢ ({{z}} ∈ ((((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c)) “_k A) ↔ ∃t ∈ A ⟪t, {{z}}⟫ ∈ (((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c)))
3		df-rex 2621	. . . . . 6 ⊢ (∃t ∈ A ⟪t, {{z}}⟫ ∈ (((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c)) ↔ ∃t(t ∈ A ∧ ⟪t, {{z}}⟫ ∈ (((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))))
4		elin 3220	. . . . . . . . . 10 ⊢ (⟪t, {{z}}⟫ ∈ (((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c)) ↔ (⟪t, {{z}}⟫ ∈ ((V ×_k V) ×_k V) ∧ ⟪t, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c)))
5	4	anbi2i 675	. . . . . . . . 9 ⊢ ((t ∈ A ∧ ⟪t, {{z}}⟫ ∈ (((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))) ↔ (t ∈ A ∧ (⟪t, {{z}}⟫ ∈ ((V ×_k V) ×_k V) ∧ ⟪t, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))))
6		an12 772	. . . . . . . . . 10 ⊢ ((t ∈ A ∧ (⟪t, {{z}}⟫ ∈ ((V ×_k V) ×_k V) ∧ ⟪t, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))) ↔ (⟪t, {{z}}⟫ ∈ ((V ×_k V) ×_k V) ∧ (t ∈ A ∧ ⟪t, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))))
7		vex 2863	. . . . . . . . . . . . . . 15 ⊢ t ∈ V
8	7, 1	opkelxpk 4249	. . . . . . . . . . . . . 14 ⊢ (⟪t, {{z}}⟫ ∈ ((V ×_k V) ×_k V) ↔ (t ∈ (V ×_k V) ∧ {{z}} ∈ V))
9	1, 8	mpbiran2 885	. . . . . . . . . . . . 13 ⊢ (⟪t, {{z}}⟫ ∈ ((V ×_k V) ×_k V) ↔ t ∈ (V ×_k V))
10		elvvk 4208	. . . . . . . . . . . . 13 ⊢ (t ∈ (V ×_k V) ↔ ∃x∃y t = ⟪x, y⟫)
11	9, 10	bitri 240	. . . . . . . . . . . 12 ⊢ (⟪t, {{z}}⟫ ∈ ((V ×_k V) ×_k V) ↔ ∃x∃y t = ⟪x, y⟫)
12	11	anbi1i 676	. . . . . . . . . . 11 ⊢ ((⟪t, {{z}}⟫ ∈ ((V ×_k V) ×_k V) ∧ (t ∈ A ∧ ⟪t, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))) ↔ (∃x∃y t = ⟪x, y⟫ ∧ (t ∈ A ∧ ⟪t, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))))
13		19.41vv 1902	. . . . . . . . . . 11 ⊢ (∃x∃y(t = ⟪x, y⟫ ∧ (t ∈ A ∧ ⟪t, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))) ↔ (∃x∃y t = ⟪x, y⟫ ∧ (t ∈ A ∧ ⟪t, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))))
14	12, 13	bitr4i 243	. . . . . . . . . 10 ⊢ ((⟪t, {{z}}⟫ ∈ ((V ×_k V) ×_k V) ∧ (t ∈ A ∧ ⟪t, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))) ↔ ∃x∃y(t = ⟪x, y⟫ ∧ (t ∈ A ∧ ⟪t, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))))
15	6, 14	bitri 240	. . . . . . . . 9 ⊢ ((t ∈ A ∧ (⟪t, {{z}}⟫ ∈ ((V ×_k V) ×_k V) ∧ ⟪t, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))) ↔ ∃x∃y(t = ⟪x, y⟫ ∧ (t ∈ A ∧ ⟪t, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))))
16	5, 15	bitri 240	. . . . . . . 8 ⊢ ((t ∈ A ∧ ⟪t, {{z}}⟫ ∈ (((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))) ↔ ∃x∃y(t = ⟪x, y⟫ ∧ (t ∈ A ∧ ⟪t, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))))
17	16	exbii 1582	. . . . . . 7 ⊢ (∃t(t ∈ A ∧ ⟪t, {{z}}⟫ ∈ (((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))) ↔ ∃t∃x∃y(t = ⟪x, y⟫ ∧ (t ∈ A ∧ ⟪t, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))))
18		exrot3 1744	. . . . . . 7 ⊢ (∃t∃x∃y(t = ⟪x, y⟫ ∧ (t ∈ A ∧ ⟪t, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))) ↔ ∃x∃y∃t(t = ⟪x, y⟫ ∧ (t ∈ A ∧ ⟪t, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))))
19	17, 18	bitri 240	. . . . . 6 ⊢ (∃t(t ∈ A ∧ ⟪t, {{z}}⟫ ∈ (((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))) ↔ ∃x∃y∃t(t = ⟪x, y⟫ ∧ (t ∈ A ∧ ⟪t, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))))
20	3, 19	bitri 240	. . . . 5 ⊢ (∃t ∈ A ⟪t, {{z}}⟫ ∈ (((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c)) ↔ ∃x∃y∃t(t = ⟪x, y⟫ ∧ (t ∈ A ∧ ⟪t, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))))
21		opkex 4114	. . . . . . . 8 ⊢ ⟪x, y⟫ ∈ V
22		eleq1 2413	. . . . . . . . 9 ⊢ (t = ⟪x, y⟫ → (t ∈ A ↔ ⟪x, y⟫ ∈ A))
23		opkeq1 4060	. . . . . . . . . 10 ⊢ (t = ⟪x, y⟫ → ⟪t, {{z}}⟫ = ⟪⟪x, y⟫, {{z}}⟫)
24	23	eleq1d 2419	. . . . . . . . 9 ⊢ (t = ⟪x, y⟫ → (⟪t, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c) ↔ ⟪⟪x, y⟫, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c)))
25	22, 24	anbi12d 691	. . . . . . . 8 ⊢ (t = ⟪x, y⟫ → ((t ∈ A ∧ ⟪t, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c)) ↔ (⟪x, y⟫ ∈ A ∧ ⟪⟪x, y⟫, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))))
26	21, 25	ceqsexv 2895	. . . . . . 7 ⊢ (∃t(t = ⟪x, y⟫ ∧ (t ∈ A ∧ ⟪t, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))) ↔ (⟪x, y⟫ ∈ A ∧ ⟪⟪x, y⟫, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c)))
27	21, 1	opkelcnvk 4251	. . . . . . . . . 10 ⊢ (⟪⟪x, y⟫, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c) ↔ ⟪{{z}}, ⟪x, y⟫⟫ ∈ ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))
28		vex 2863	. . . . . . . . . . 11 ⊢ z ∈ V
29		vex 2863	. . . . . . . . . . 11 ⊢ x ∈ V
30		vex 2863	. . . . . . . . . . 11 ⊢ y ∈ V
31	28, 29, 30	setconslem3 4734	. . . . . . . . . 10 ⊢ (⟪{{z}}, ⟪x, y⟫⟫ ∈ ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c) ↔ z = ⟨x, y⟩)
32	27, 31	bitri 240	. . . . . . . . 9 ⊢ (⟪⟪x, y⟫, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c) ↔ z = ⟨x, y⟩)
33	32	anbi2i 675	. . . . . . . 8 ⊢ ((⟪x, y⟫ ∈ A ∧ ⟪⟪x, y⟫, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c)) ↔ (⟪x, y⟫ ∈ A ∧ z = ⟨x, y⟩))
34		ancom 437	. . . . . . . 8 ⊢ ((⟪x, y⟫ ∈ A ∧ z = ⟨x, y⟩) ↔ (z = ⟨x, y⟩ ∧ ⟪x, y⟫ ∈ A))
35	33, 34	bitri 240	. . . . . . 7 ⊢ ((⟪x, y⟫ ∈ A ∧ ⟪⟪x, y⟫, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c)) ↔ (z = ⟨x, y⟩ ∧ ⟪x, y⟫ ∈ A))
36	26, 35	bitri 240	. . . . . 6 ⊢ (∃t(t = ⟪x, y⟫ ∧ (t ∈ A ∧ ⟪t, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))) ↔ (z = ⟨x, y⟩ ∧ ⟪x, y⟫ ∈ A))
37	36	2exbii 1583	. . . . 5 ⊢ (∃x∃y∃t(t = ⟪x, y⟫ ∧ (t ∈ A ∧ ⟪t, {{z}}⟫ ∈ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c))) ↔ ∃x∃y(z = ⟨x, y⟩ ∧ ⟪x, y⟫ ∈ A))
38	20, 37	bitri 240	. . . 4 ⊢ (∃t ∈ A ⟪t, {{z}}⟫ ∈ (((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c)) ↔ ∃x∃y(z = ⟨x, y⟩ ∧ ⟪x, y⟫ ∈ A))
39	2, 38	bitri 240	. . 3 ⊢ ({{z}} ∈ ((((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c)) “_k A) ↔ ∃x∃y(z = ⟨x, y⟩ ∧ ⟪x, y⟫ ∈ A))
40	28	eluni1 4174	. . . 4 ⊢ (z ∈ ⋃₁⋃₁((((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c)) “_k A) ↔ {z} ∈ ⋃₁((((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c)) “_k A))
41		snex 4112	. . . . 5 ⊢ {z} ∈ V
42	41	eluni1 4174	. . . 4 ⊢ ({z} ∈ ⋃₁((((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c)) “_k A) ↔ {{z}} ∈ ((((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c)) “_k A))
43	40, 42	bitri 240	. . 3 ⊢ (z ∈ ⋃₁⋃₁((((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c)) “_k A) ↔ {{z}} ∈ ((((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c)) “_k A))
44		elopab 4697	. . 3 ⊢ (z ∈ {⟨x, y⟩ ∣ ⟪x, y⟫ ∈ A} ↔ ∃x∃y(z = ⟨x, y⟩ ∧ ⟪x, y⟫ ∈ A))
45	39, 43, 44	3bitr4i 268	. 2 ⊢ (z ∈ ⋃₁⋃₁((((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c)) “_k A) ↔ z ∈ {⟨x, y⟩ ∣ ⟪x, y⟫ ∈ A})
46	45	eqriv 2350	1 ⊢ ⋃₁⋃₁((((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_k ⊕ Ins2_k ( Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c)) “_k A) = {⟨x, y⟩ ∣ ⟪x, y⟫ ∈ A}

Colors of variables: wff setvar class

Syntax hints: ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 ∃wrex 2616 Vcvv 2860 ∼ ccompl 3206 ∖ cdif 3207 ∪ cun 3208 ∩ cin 3209 ⊕ csymdif 3210 {csn 3738 ⟪copk 4058 ⋃₁cuni1 4134 1_cc1c 4135 ℘₁cpw1 4136 ×_k cxpk 4175 ^◡_kccnvk 4176 Ins2_k cins2k 4177 Ins3_k cins3k 4178 “_k cimak 4180 ∘_k ccomk 4181 SI_k csik 4182 Image_kcimagek 4183 S_k cssetk 4184 I_k cidk 4185 Nn cnnc 4374 0_cc0c 4375 ⟨cop 4562 {copab 4623

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-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-addc 4379 df-nnc 4380 df-phi 4566 df-op 4567 df-opab 4624

This theorem is referenced by: df1st2 4739 dfsset2 4744 dfco1 4749 dfsi2 4752

W3C validator