Theorem nchoicelem18 6307

Description: Lemma for nchoice 6309. Set up stratification for nchoicelem19 6308. (Contributed by SF, 20-Mar-2015.)

Assertion

Ref	Expression
nchoicelem18	|- {x | ( Spac ` x) e. Fin } e. _V

Proof of Theorem nchoicelem18

Dummy variables c p q are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm2.1 406	. . . 4 |- (-. x e. NC \/ x e. NC )
2		fnspac 6284	. . . . . . . . . . 11 |- Spac Fn NC
3		fndm 5183	. . . . . . . . . . 11 |- ( Spac Fn NC -> dom Spac = NC )
4	2, 3	ax-mp 5	. . . . . . . . . 10 |- dom Spac = NC
5	4	eleq2i 2417	. . . . . . . . 9 |- (x e. dom Spac <-> x e. NC )
6		ndmfv 5350	. . . . . . . . 9 |- (-. x e. dom Spac -> ( Spac ` x) = (/))
7	5, 6	sylnbir 298	. . . . . . . 8 |- (-. x e. NC -> ( Spac ` x) = (/))
8		0fin 4424	. . . . . . . 8 |- (/) e. Fin
9	7, 8	syl6eqel 2441	. . . . . . 7 |- (-. x e. NC -> ( Spac ` x) e. Fin )
10	9	pm4.71i 613	. . . . . 6 |- (-. x e. NC <-> (-. x e. NC /\ ( Spac ` x) e. Fin ))
11	10	orbi1i 506	. . . . 5 |- ((-. x e. NC \/ (x e. NC /\ ( Spac ` x) e. Fin )) <-> ((-. x e. NC /\ ( Spac ` x) e. Fin ) \/ (x e. NC /\ ( Spac ` x) e. Fin )))
12		elun 3221	. . . . . 6 |- (x e. ( ∼ NC u. ( NC i^i ⋃1( ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c)" Fin ))) <-> (x e. ∼ NC \/ x e. ( NC i^i ⋃1( ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c)" Fin ))))
13		vex 2863	. . . . . . . 8 |- x e. _V
14	13	elcompl 3226	. . . . . . 7 |- (x e. ∼ NC <-> -. x e. NC )
15		elin 3220	. . . . . . . 8 |- (x e. ( NC i^i ⋃1( ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c)" Fin )) <-> (x e. NC /\ x e. ⋃1( ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c)" Fin )))
16		spacval 6283	. . . . . . . . . . 11 |- (x e. NC -> ( Spac ` x) = Clos1 ({x}, {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))
17	16	eleq1d 2419	. . . . . . . . . 10 |- (x e. NC -> (( Spac ` x) e. Fin <-> Clos1 ({x}, {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}) e. Fin ))
18	13	eluni1 4174	. . . . . . . . . . 11 |- (x e. ⋃1( ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c)" Fin ) <-> {x} e. ( ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c)" Fin ))
19		df-br 4641	. . . . . . . . . . . . . 14 |- (c ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c){x} <-> <.c, {x}>. e. ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c))
20		spacvallem1 6282	. . . . . . . . . . . . . . 15 |- {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))} e. _V
21		snex 4112	. . . . . . . . . . . . . . 15 |- {x} e. _V
22	20, 21	nchoicelem10 6299	. . . . . . . . . . . . . 14 |- (<.c, {x}>. e. ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c) <-> c = Clos1 ({x}, {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))
23	19, 22	bitri 240	. . . . . . . . . . . . 13 |- (c ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c){x} <-> c = Clos1 ({x}, {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))
24	23	rexbii 2640	. . . . . . . . . . . 12 |- (E.c e. Fin c ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c){x} <-> E.c e. Fin c = Clos1 ({x}, {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))
25		elima 4755	. . . . . . . . . . . 12 |- ({x} e. ( ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c)" Fin ) <-> E.c e. Fin c ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c){x})
26		risset 2662	. . . . . . . . . . . 12 |- ( Clos1 ({x}, {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}) e. Fin <-> E.c e. Fin c = Clos1 ({x}, {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))
27	24, 25, 26	3bitr4i 268	. . . . . . . . . . 11 |- ({x} e. ( ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c)" Fin ) <-> Clos1 ({x}, {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}) e. Fin )
28	18, 27	bitri 240	. . . . . . . . . 10 |- (x e. ⋃1( ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c)" Fin ) <-> Clos1 ({x}, {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}) e. Fin )
29	17, 28	syl6rbbr 255	. . . . . . . . 9 |- (x e. NC -> (x e. ⋃1( ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c)" Fin ) <-> ( Spac ` x) e. Fin ))
30	29	pm5.32i 618	. . . . . . . 8 |- ((x e. NC /\ x e. ⋃1( ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c)" Fin )) <-> (x e. NC /\ ( Spac ` x) e. Fin ))
31	15, 30	bitri 240	. . . . . . 7 |- (x e. ( NC i^i ⋃1( ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c)" Fin )) <-> (x e. NC /\ ( Spac ` x) e. Fin ))
32	14, 31	orbi12i 507	. . . . . 6 |- ((x e. ∼ NC \/ x e. ( NC i^i ⋃1( ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c)" Fin ))) <-> (-. x e. NC \/ (x e. NC /\ ( Spac ` x) e. Fin )))
33	12, 32	bitri 240	. . . . 5 |- (x e. ( ∼ NC u. ( NC i^i ⋃1( ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c)" Fin ))) <-> (-. x e. NC \/ (x e. NC /\ ( Spac ` x) e. Fin )))
34		andir 838	. . . . 5 |- (((-. x e. NC \/ x e. NC ) /\ ( Spac ` x) e. Fin ) <-> ((-. x e. NC /\ ( Spac ` x) e. Fin ) \/ (x e. NC /\ ( Spac ` x) e. Fin )))
35	11, 33, 34	3bitr4i 268	. . . 4 |- (x e. ( ∼ NC u. ( NC i^i ⋃1( ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c)" Fin ))) <-> ((-. x e. NC \/ x e. NC ) /\ ( Spac ` x) e. Fin ))
36	1, 35	mpbiran 884	. . 3 |- (x e. ( ∼ NC u. ( NC i^i ⋃1( ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c)" Fin ))) <-> ( Spac ` x) e. Fin )
37	36	abbi2i 2465	. 2 |- ( ∼ NC u. ( NC i^i ⋃1( ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c)" Fin ))) = {x | ( Spac ` x) e. Fin }
38		ncsex 6112	. . . 4 |- NC e. _V
39	38	complex 4105	. . 3 |- ∼ NC e. _V
40		ssetex 4745	. . . . . . . . . 10 |- S e. _V
41	40	ins3ex 5799	. . . . . . . . 9 |- Ins3 S e. _V
42	40	complex 4105	. . . . . . . . . . . . . 14 |- ∼ S e. _V
43	42	cnvex 5103	. . . . . . . . . . . . 13 |- `' ∼ S e. _V
44	40	cnvex 5103	. . . . . . . . . . . . . 14 |- `' S e. _V
45	20	imageex 5802	. . . . . . . . . . . . . . . 16 |- Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))} e. _V
46	40, 45	coex 4751	. . . . . . . . . . . . . . 15 |- ( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}) e. _V
47	46	fixex 5790	. . . . . . . . . . . . . 14 |- Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}) e. _V
48	44, 47	resex 5118	. . . . . . . . . . . . 13 |- (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))})) e. _V
49	43, 48	txpex 5786	. . . . . . . . . . . 12 |- (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))) e. _V
50	49	rnex 5108	. . . . . . . . . . 11 |- ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))) e. _V
51	50	complex 4105	. . . . . . . . . 10 |- ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))) e. _V
52	51	ins2ex 5798	. . . . . . . . 9 |- Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))) e. _V
53	41, 52	symdifex 4109	. . . . . . . 8 |- ( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))})))) e. _V
54		1cex 4143	. . . . . . . 8 |- 1c e. _V
55	53, 54	imaex 4748	. . . . . . 7 |- (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c) e. _V
56	55	complex 4105	. . . . . 6 |- ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c) e. _V
57		finex 4398	. . . . . 6 |- Fin e. _V
58	56, 57	imaex 4748	. . . . 5 |- ( ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c)" Fin ) e. _V
59	58	uni1ex 4294	. . . 4 |- ⋃1( ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c)" Fin ) e. _V
60	38, 59	inex 4106	. . 3 |- ( NC i^i ⋃1( ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c)" Fin )) e. _V
61	39, 60	unex 4107	. 2 |- ( ∼ NC u. ( NC i^i ⋃1( ∼ (( Ins3 S (+) Ins2 ∼ ran (`' ∼ S (x) (`' S |` Fix( S o. Image{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))))"1c)" Fin ))) e. _V
62	37, 61	eqeltrri 2424	1 |- {x | ( Spac ` x) e. Fin } e. _V

Syntax hints:  -. wn 3   \/ wo 357   /\ wa 358   /\ w3a 934   = wceq 1642   e. wcel 1710  {cab 2339  E.wrex 2616  _Vcvv 2860   ∼ ccompl 3206   u. cun 3208   i^i cin 3209   (+) csymdif 3210  (/)c0 3551  {csn 3738  ⋃₁cuni1 4134  1_cc1c 4135   Fin cfin 4377  <.cop 4562  {copab 4623   class class class wbr 4640   S csset 4720   o. ccom 4722  "cima 4723  `'ccnv 4772  dom cdm 4773  ran crn 4774   |` cres 4775   Fn wfn 4777  ` cfv 4782  (class class class)co 5526   (x) ctxp 5736  Fixcfix 5740   Ins2 cins2 5750   Ins3 cins3 5752  Imagecimage 5754   Clos1 cclos1 5873   NC cncs 6089  2_cc2c 6095   ↑_c cce 6097   Sp_ac cspac 6274

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-13 1712  ax-14 1714  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-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  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-pss 3262  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-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-fix 5741  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-pw1fn 5767  df-fullfun 5769  df-clos1 5874  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-map 6002  df-en 6030  df-ncs 6099  df-nc 6102  df-2c 6105  df-ce 6107  df-spac 6275

This theorem is referenced by:  nchoicelem19  6308