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Theorem srelkex 4526

Description: The expression at the core of srelk 4525 exists. (Contributed by SF, 30-Jan-2015.)

**Assertion**
Ref	Expression
srelkex	⊢ (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁1_c) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁1_c)) ∈ V

**Proof of Theorem srelkex**
Step	Hyp	Ref	Expression
1		nncex 4397	. . 3 ⊢ Nn ∈ V
2	1, 1	xpkex 4290	. 2 ⊢ ( Nn ×_k Nn ) ∈ V
3		1cex 4143	. . . . . . . . . . . 12 ⊢ 1_c ∈ V
4	3	pwex 4330	. . . . . . . . . . 11 ⊢ ℘1_c ∈ V
5		vvex 4110	. . . . . . . . . . 11 ⊢ V ∈ V
6	4, 5	xpkex 4290	. . . . . . . . . 10 ⊢ (℘1_c ×_k V) ∈ V
7		ssetkex 4295	. . . . . . . . . . . . 13 ⊢ S_k ∈ V
8	7	ins3kex 4309	. . . . . . . . . . . 12 ⊢ Ins3_k S_k ∈ V
9	7	sikex 4298	. . . . . . . . . . . . 13 ⊢ SI_k S_k ∈ V
10	9	ins2kex 4308	. . . . . . . . . . . 12 ⊢ Ins2_k SI_k S_k ∈ V
11	8, 10	symdifex 4109	. . . . . . . . . . 11 ⊢ ( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) ∈ V
12	3	pw1ex 4304	. . . . . . . . . . . . 13 ⊢ ℘₁1_c ∈ V
13	12	pw1ex 4304	. . . . . . . . . . . 12 ⊢ ℘₁℘₁1_c ∈ V
14	13	pw1ex 4304	. . . . . . . . . . 11 ⊢ ℘₁℘₁℘₁1_c ∈ V
15	11, 14	imakex 4301	. . . . . . . . . 10 ⊢ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c) ∈ V
16	6, 15	difex 4108	. . . . . . . . 9 ⊢ ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∈ V
17	16	sikex 4298	. . . . . . . 8 ⊢ SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∈ V
18	17	ins3kex 4309	. . . . . . 7 ⊢ Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∈ V
19	7	ins2kex 4308	. . . . . . 7 ⊢ Ins2_k S_k ∈ V
20	18, 19	inex 4106	. . . . . 6 ⊢ ( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) ∈ V
21	20, 13	imakex 4301	. . . . 5 ⊢ (( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∈ V
22	21	ins3kex 4309	. . . 4 ⊢ Ins3_k (( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∈ V
23	11, 13	imakex 4301	. . . . . . . . . 10 ⊢ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁1_c) ∈ V
24	23	complex 4105	. . . . . . . . 9 ⊢ ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁1_c) ∈ V
25	24	sikex 4298	. . . . . . . 8 ⊢ SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁1_c) ∈ V
26	25	ins3kex 4309	. . . . . . 7 ⊢ Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁1_c) ∈ V
27	26, 19	inex 4106	. . . . . 6 ⊢ ( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁1_c) ∩ Ins2_k S_k ) ∈ V
28	27, 13	imakex 4301	. . . . 5 ⊢ (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁1_c) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∈ V
29	28	ins2kex 4308	. . . 4 ⊢ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁1_c) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∈ V
30	22, 29	inex 4106	. . 3 ⊢ ( Ins3_k (( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁1_c) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c)) ∈ V
31	30, 14	imakex 4301	. 2 ⊢ (( Ins3_k (( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁1_c) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁1_c) ∈ V
32	2, 31	inex 4106	1 ⊢ (( Nn ×_k Nn ) ∩ (( Ins3_k (( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∩ Ins2_k (( Ins3_k SI_k ∼ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁1_c) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁1_c)) ∈ V

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∈ wcel 1710 Vcvv 2860 ∼ ccompl 3206 ∖ cdif 3207 ∩ cin 3209 ⊕ csymdif 3210 ℘cpw 3723 1_cc1c 4135 ℘₁cpw1 4136 ×_k cxpk 4175 Ins2_k cins2k 4177 Ins3_k cins3k 4178 “_k cimak 4180 SI_k csik 4182 S_k cssetk 4184 Nn cnnc 4374

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-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-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-addc 4379 df-nnc 4380

This theorem is used by: sfintfinlem1 4532 spfinex 4538

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