df1st2 - New Foundations Explorer

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Theorem df1st2 4739

Description: Express the

function via the set construction functions. (Contributed by SF, 4-Feb-2015.)

**Assertion**
Ref	Expression
df1st2	⋃₁⋃₁ _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ 1_c

Proof of Theorem df1st2 Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opkex 4114	. . . . . 6
2	1	elimak 4260	. . . . 5 ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ 1_c ₁ 1_c ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c
3		df-rex 2621	. . . . . 6 ₁ 1_c ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c ₁ 1_c $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c
4		elpw11c 4148	. . . . . . . . . 10 ₁ 1_c
5	4	anbi1i 676	. . . . . . . . 9 ₁ 1_c $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c
6		19.41v 1901	. . . . . . . . 9 $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c
7	5, 6	bitr4i 243	. . . . . . . 8 ₁ 1_c $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c
8	7	exbii 1582	. . . . . . 7 ₁ 1_c $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c
9		excom 1741	. . . . . . 7 $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c
10	8, 9	bitr4i 243	. . . . . 6 ₁ 1_c $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c
11	3, 10	bitri 240	. . . . 5 ₁ 1_c ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c
12	2, 11	bitri 240	. . . 4 ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ 1_c $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c
13		snex 4112	. . . . . . 7
14		opkeq1 4060	. . . . . . . 8
15	14	eleq1d 2419	. . . . . . 7 ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c
16	13, 15	ceqsexv 2895	. . . . . 6 $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c
17		vex 2863	. . . . . . 7
18		vex 2863	. . . . . . 7
19		vex 2863	. . . . . . 7
20	17, 18, 19	setconslem7 4738	. . . . . 6 ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c
21	16, 20	bitri 240	. . . . 5 $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c
22	21	exbii 1582	. . . 4 $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c
23	12, 22	bitri 240	. . 3 ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ 1_c
24	23	opabbii 4627	. 2 ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ 1_c
25		setconslem4 4735	. 2 ⋃₁⋃₁ _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ 1_c ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ 1_c
26		df-1st 4724	. 2
27	24, 25, 26	3eqtr4ri 2384	1 ⋃₁⋃₁ _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins3_k SI_k SI_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ 1_c

Colors of variables: wff setvar class

Syntax hints: $/\$ wa 358

wex 1541

wceq 1642

wcel 1710

wrex 2616

cvv 2860 ∼ ccompl 3206

cdif 3207

cun 3208

cin 3209

csymdif 3210

csn 3738

copk 4058 ⋃₁cuni1 4134 1_cc1c 4135

₁ cpw1 4136

_k cxpk 4175

_kccnvk 4176 Ins2_k cins2k 4177 Ins3_k cins3k 4178

_kcimak 4180

_k ccomk 4181 SI_k csik 4182 Image_kcimagek 4183 S_k cssetk 4184

_k cidk 4185 Nn cnnc 4374 0_cc0c 4375

cop 4562

copab 4623

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

W3C validator