df1st2 - New Foundations Explorer

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Theorem df1st2 4739

Description: Express the 1^st function via the set construction functions. (Contributed by SF, 4-Feb-2015.)

**Assertion**
Ref	Expression
df1st2	⊢ 1^st = ⋃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 ( ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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 ℘₁1_c))

Proof of Theorem df1st2 Dummy variables x y z t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opkex 4114	. . . . . 6 ⊢ ⟪x, y⟫ ∈ V
2	1	elimak 4260	. . . . 5 ⊢ (⟪x, y⟫ ∈ ( ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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 ℘₁1_c) ↔ ∃t ∈ ℘₁ 1_c⟪t, ⟪x, y⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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 ∈ ℘₁ 1_c⟪t, ⟪x, y⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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 ∈ ℘₁1_c ∧ ⟪t, ⟪x, y⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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		elpw11c 4148	. . . . . . . . . 10 ⊢ (t ∈ ℘₁1_c ↔ ∃z t = {{z}})
5	4	anbi1i 676	. . . . . . . . 9 ⊢ ((t ∈ ℘₁1_c ∧ ⟪t, ⟪x, y⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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 t = {{z}} ∧ ⟪t, ⟪x, y⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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		19.41v 1901	. . . . . . . . 9 ⊢ (∃z(t = {{z}} ∧ ⟪t, ⟪x, y⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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 t = {{z}} ∧ ⟪t, ⟪x, y⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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	5, 6	bitr4i 243	. . . . . . . 8 ⊢ ((t ∈ ℘₁1_c ∧ ⟪t, ⟪x, y⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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(t = {{z}} ∧ ⟪t, ⟪x, y⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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)))
8	7	exbii 1582	. . . . . . 7 ⊢ (∃t(t ∈ ℘₁1_c ∧ ⟪t, ⟪x, y⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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(t = {{z}} ∧ ⟪t, ⟪x, y⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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)))
9		excom 1741	. . . . . . 7 ⊢ (∃z∃t(t = {{z}} ∧ ⟪t, ⟪x, y⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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(t = {{z}} ∧ ⟪t, ⟪x, y⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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)))
10	8, 9	bitr4i 243	. . . . . 6 ⊢ (∃t(t ∈ ℘₁1_c ∧ ⟪t, ⟪x, y⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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∃t(t = {{z}} ∧ ⟪t, ⟪x, y⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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)))
11	3, 10	bitri 240	. . . . 5 ⊢ (∃t ∈ ℘₁ 1_c⟪t, ⟪x, y⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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∃t(t = {{z}} ∧ ⟪t, ⟪x, y⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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)))
12	2, 11	bitri 240	. . . 4 ⊢ (⟪x, y⟫ ∈ ( ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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 ℘₁1_c) ↔ ∃z∃t(t = {{z}} ∧ ⟪t, ⟪x, y⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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		snex 4112	. . . . . . 7 ⊢ {{z}} ∈ V
14		opkeq1 4060	. . . . . . . 8 ⊢ (t = {{z}} → ⟪t, ⟪x, y⟫⟫ = ⟪{{z}}, ⟪x, y⟫⟫)
15	14	eleq1d 2419	. . . . . . 7 ⊢ (t = {{z}} → (⟪t, ⟪x, y⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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	13, 15	ceqsexv 2895	. . . . . 6 ⊢ (∃t(t = {{z}} ∧ ⟪t, ⟪x, y⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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		vex 2863	. . . . . . 7 ⊢ x ∈ V
18		vex 2863	. . . . . . 7 ⊢ y ∈ V
19		vex 2863	. . . . . . 7 ⊢ z ∈ V
20	17, 18, 19	setconslem7 4738	. . . . . 6 ⊢ (⟪{{z}}, ⟪x, y⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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⟩)
21	16, 20	bitri 240	. . . . 5 ⊢ (∃t(t = {{z}} ∧ ⟪t, ⟪x, y⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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⟩)
22	21	exbii 1582	. . . 4 ⊢ (∃z∃t(t = {{z}} ∧ ⟪t, ⟪x, y⟫⟫ ∈ ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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, z⟩)
23	12, 22	bitri 240	. . 3 ⊢ (⟪x, y⟫ ∈ ( ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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 ℘₁1_c) ↔ ∃z x = ⟨y, z⟩)
24	23	opabbii 4627	. 2 ⊢ {⟨x, y⟩ ∣ ⟪x, y⟫ ∈ ( ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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 ℘₁1_c)} = {⟨x, y⟩ ∣ ∃z x = ⟨y, z⟩}
25		setconslem4 4735	. 2 ⊢ ⋃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 ( ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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 ℘₁1_c)) = {⟨x, y⟩ ∣ ⟪x, y⟫ ∈ ( ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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 ℘₁1_c)}
26		df-1st 4724	. 2 ⊢ 1^st = {⟨x, y⟩ ∣ ∃z x = ⟨y, z⟩}
27	24, 25, 26	3eqtr4ri 2384	1 ⊢ 1^st = ⋃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 ( ∼ (( Ins2_k Ins3_k S_k ⊕ ( Ins2_k Ins2_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)))) ∪ Ins3_k SI_k SI_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 ℘₁1_c))

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 1^st c1st 4718

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 df-1st 4724

This theorem is referenced by: 1stex 4740

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