tfinrelkex - New Foundations Explorer

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Theorem tfinrelkex 4488

Description: The expression at the core of eqtfinrelk 4487 exists. (Contributed by SF, 30-Jan-2015.)

**Assertion**
Ref	Expression
tfinrelkex	⊢ (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ 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)) “_k ℘₁℘₁℘₁1_c)) ⊕ Ins3_k I_k ) “_k ℘₁1_c)) “_k ℘₁1_c)) “_k ℘₁℘₁1_c) ∖ ({{∅}} ×_k V))) ∈ V

**Proof of Theorem tfinrelkex**
Step	Hyp	Ref	Expression
1		snex 4112	. . 3 ⊢ {{∅}} ∈ V
2		snex 4112	. . 3 ⊢ {∅} ∈ V
3	1, 2	xpkex 4290	. 2 ⊢ ({{∅}} ×_k {∅}) ∈ V
4		ssetkex 4295	. . . . . . 7 ⊢ S_k ∈ V
5	4	ins2kex 4308	. . . . . 6 ⊢ Ins2_k S_k ∈ V
6	4	cnvkex 4288	. . . . . . . . . 10 ⊢ ^◡_k S_k ∈ V
7	6	ins3kex 4309	. . . . . . . . 9 ⊢ Ins3_k ^◡_k S_k ∈ V
8		nncex 4397	. . . . . . . . . . . . . . 15 ⊢ Nn ∈ V
9		vvex 4110	. . . . . . . . . . . . . . 15 ⊢ V ∈ V
10	8, 9	xpkex 4290	. . . . . . . . . . . . . 14 ⊢ ( Nn ×_k V) ∈ V
11	4	sikex 4298	. . . . . . . . . . . . . . . . 17 ⊢ SI_k S_k ∈ V
12	11	ins2kex 4308	. . . . . . . . . . . . . . . 16 ⊢ Ins2_k SI_k S_k ∈ V
13		1cex 4143	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1_c ∈ V
14	13	pwex 4330	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℘1_c ∈ V
15	14, 9	xpkex 4290	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (℘1_c ×_k V) ∈ V
16	4	ins3kex 4309	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ins3_k S_k ∈ V
17	16, 12	symdifex 4109	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) ∈ V
18	13	pw1ex 4304	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ℘₁1_c ∈ V
19	18	pw1ex 4304	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ℘₁℘₁1_c ∈ V
20	19	pw1ex 4304	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℘₁℘₁℘₁1_c ∈ V
21	17, 20	imakex 4301	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c) ∈ V
22	15, 21	difex 4108	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∈ V
23	22	sikex 4298	. . . . . . . . . . . . . . . . . . . 20 ⊢ SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∈ V
24	23	ins3kex 4309	. . . . . . . . . . . . . . . . . . 19 ⊢ Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∈ V
25	24, 5	inex 4106	. . . . . . . . . . . . . . . . . 18 ⊢ ( 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
26	25, 19	imakex 4301	. . . . . . . . . . . . . . . . 17 ⊢ (( 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
27	26	ins3kex 4309	. . . . . . . . . . . . . . . 16 ⊢ 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
28	12, 27	inex 4106	. . . . . . . . . . . . . . 15 ⊢ ( Ins2_k SI_k S_k ∩ 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
29	28, 20	imakex 4301	. . . . . . . . . . . . . 14 ⊢ (( Ins2_k SI_k S_k ∩ 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)) “_k ℘₁℘₁℘₁1_c) ∈ V
30	10, 29	inex 4106	. . . . . . . . . . . . 13 ⊢ (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ 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)) “_k ℘₁℘₁℘₁1_c)) ∈ V
31	30	ins2kex 4308	. . . . . . . . . . . 12 ⊢ Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ 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)) “_k ℘₁℘₁℘₁1_c)) ∈ V
32		idkex 4315	. . . . . . . . . . . . 13 ⊢ I_k ∈ V
33	32	ins3kex 4309	. . . . . . . . . . . 12 ⊢ Ins3_k I_k ∈ V
34	31, 33	symdifex 4109	. . . . . . . . . . 11 ⊢ ( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ 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)) “_k ℘₁℘₁℘₁1_c)) ⊕ Ins3_k I_k ) ∈ V
35	34, 18	imakex 4301	. . . . . . . . . 10 ⊢ (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ 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)) “_k ℘₁℘₁℘₁1_c)) ⊕ Ins3_k I_k ) “_k ℘₁1_c) ∈ V
36	35	ins2kex 4308	. . . . . . . . 9 ⊢ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ 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)) “_k ℘₁℘₁℘₁1_c)) ⊕ Ins3_k I_k ) “_k ℘₁1_c) ∈ V
37	7, 36	difex 4108	. . . . . . . 8 ⊢ ( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ 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)) “_k ℘₁℘₁℘₁1_c)) ⊕ Ins3_k I_k ) “_k ℘₁1_c)) ∈ V
38	37, 18	imakex 4301	. . . . . . 7 ⊢ (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ 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)) “_k ℘₁℘₁℘₁1_c)) ⊕ Ins3_k I_k ) “_k ℘₁1_c)) “_k ℘₁1_c) ∈ V
39	38	ins3kex 4309	. . . . . 6 ⊢ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ 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)) “_k ℘₁℘₁℘₁1_c)) ⊕ Ins3_k I_k ) “_k ℘₁1_c)) “_k ℘₁1_c) ∈ V
40	5, 39	symdifex 4109	. . . . 5 ⊢ ( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ 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)) “_k ℘₁℘₁℘₁1_c)) ⊕ Ins3_k I_k ) “_k ℘₁1_c)) “_k ℘₁1_c)) ∈ V
41	40, 19	imakex 4301	. . . 4 ⊢ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ 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)) “_k ℘₁℘₁℘₁1_c)) ⊕ Ins3_k I_k ) “_k ℘₁1_c)) “_k ℘₁1_c)) “_k ℘₁℘₁1_c) ∈ V
42	41	complex 4105	. . 3 ⊢ ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ 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)) “_k ℘₁℘₁℘₁1_c)) ⊕ Ins3_k I_k ) “_k ℘₁1_c)) “_k ℘₁1_c)) “_k ℘₁℘₁1_c) ∈ V
43	1, 9	xpkex 4290	. . 3 ⊢ ({{∅}} ×_k V) ∈ V
44	42, 43	difex 4108	. 2 ⊢ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ 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)) “_k ℘₁℘₁℘₁1_c)) ⊕ Ins3_k I_k ) “_k ℘₁1_c)) “_k ℘₁1_c)) “_k ℘₁℘₁1_c) ∖ ({{∅}} ×_k V)) ∈ V
45	3, 44	unex 4107	1 ⊢ (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ 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)) “_k ℘₁℘₁℘₁1_c)) ⊕ Ins3_k I_k ) “_k ℘₁1_c)) “_k ℘₁1_c)) “_k ℘₁℘₁1_c) ∖ ({{∅}} ×_k V))) ∈ V

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∈ wcel 1710 Vcvv 2860 ∼ ccompl 3206 ∖ cdif 3207 ∪ cun 3208 ∩ cin 3209 ⊕ csymdif 3210 ∅c0 3551 ℘cpw 3723 {csn 3738 1_cc1c 4135 ℘₁cpw1 4136 ×_k cxpk 4175 ^◡_kccnvk 4176 Ins2_k cins2k 4177 Ins3_k cins3k 4178 “_k cimak 4180 SI_k csik 4182 S_k cssetk 4184 I_k cidk 4185 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-idk 4196 df-addc 4379 df-nnc 4380

This theorem is used by: sfintfinlem1 4532 tfinnnlem1 4534 vfinspss 4552 vfinspclt 4553 vfinncsp 4555

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