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Theorem nchoicelem18 6306
 Description: Lemma for nchoice 6308. Set up stratification for nchoicelem19 6307. (Contributed by SF, 20-Mar-2015.)
Assertion
Ref Expression
nchoicelem18 {x ( Spacx) Fin } V

Proof of Theorem nchoicelem18
Dummy variables c p q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pm2.1 406 . . . 4 x NC x NC )
2 fnspac 6283 . . . . . . . . . . 11 Spac Fn NC
3 fndm 5182 . . . . . . . . . . 11 ( Spac Fn NC → dom Spac = NC )
42, 3ax-mp 8 . . . . . . . . . 10 dom Spac = NC
54eleq2i 2417 . . . . . . . . 9 (x dom Spacx NC )
6 ndmfv 5349 . . . . . . . . 9 x dom Spac → ( Spacx) = )
75, 6sylnbir 298 . . . . . . . 8 x NC → ( Spacx) = )
8 0fin 4423 . . . . . . . 8 Fin
97, 8syl6eqel 2441 . . . . . . 7 x NC → ( Spacx) Fin )
109pm4.71i 613 . . . . . 6 x NC ↔ (¬ x NC ( Spacx) Fin ))
1110orbi1i 506 . . . . 5 ((¬ x NC (x NC ( Spacx) Fin )) ↔ ((¬ x NC ( Spacx) Fin ) (x NC ( Spacx) Fin )))
12 elun 3220 . . . . . 6 (x ( ∼ NC ∪ ( NC ∩ ⋃1( ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c) “ Fin ))) ↔ (x NC x ( NC ∩ ⋃1( ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c) “ Fin ))))
13 vex 2862 . . . . . . . 8 x V
1413elcompl 3225 . . . . . . 7 (x NC ↔ ¬ x NC )
15 elin 3219 . . . . . . . 8 (x ( NC ∩ ⋃1( ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c) “ Fin )) ↔ (x NC x 1( ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c) “ Fin )))
16 spacval 6282 . . . . . . . . . . 11 (x NC → ( Spacx) = Clos1 ({x}, {p, q (p NC q NC q = (2cc p))}))
1716eleq1d 2419 . . . . . . . . . 10 (x NC → (( Spacx) Fin Clos1 ({x}, {p, q (p NC q NC q = (2cc p))}) Fin ))
1813eluni1 4173 . . . . . . . . . . 11 (x 1( ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c) “ Fin ) ↔ {x} ( ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c) “ Fin ))
19 df-br 4640 . . . . . . . . . . . . . 14 (c ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c){x} ↔ c, {x} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c))
20 spacvallem1 6281 . . . . . . . . . . . . . . 15 {p, q (p NC q NC q = (2cc p))} V
21 snex 4111 . . . . . . . . . . . . . . 15 {x} V
2220, 21nchoicelem10 6298 . . . . . . . . . . . . . 14 (c, {x} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c) ↔ c = Clos1 ({x}, {p, q (p NC q NC q = (2cc p))}))
2319, 22bitri 240 . . . . . . . . . . . . 13 (c ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c){x} ↔ c = Clos1 ({x}, {p, q (p NC q NC q = (2cc p))}))
2423rexbii 2639 . . . . . . . . . . . 12 (c Fin c ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c){x} ↔ c Fin c = Clos1 ({x}, {p, q (p NC q NC q = (2cc p))}))
25 elima 4754 . . . . . . . . . . . 12 ({x} ( ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c) “ Fin ) ↔ c Fin c ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c){x})
26 risset 2661 . . . . . . . . . . . 12 ( Clos1 ({x}, {p, q (p NC q NC q = (2cc p))}) Finc Fin c = Clos1 ({x}, {p, q (p NC q NC q = (2cc p))}))
2724, 25, 263bitr4i 268 . . . . . . . . . . 11 ({x} ( ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c) “ Fin ) ↔ Clos1 ({x}, {p, q (p NC q NC q = (2cc p))}) Fin )
2818, 27bitri 240 . . . . . . . . . 10 (x 1( ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c) “ Fin ) ↔ Clos1 ({x}, {p, q (p NC q NC q = (2cc p))}) Fin )
2917, 28syl6rbbr 255 . . . . . . . . 9 (x NC → (x 1( ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c) “ Fin ) ↔ ( Spacx) Fin ))
3029pm5.32i 618 . . . . . . . 8 ((x NC x 1( ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c) “ Fin )) ↔ (x NC ( Spacx) Fin ))
3115, 30bitri 240 . . . . . . 7 (x ( NC ∩ ⋃1( ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c) “ Fin )) ↔ (x NC ( Spacx) Fin ))
3214, 31orbi12i 507 . . . . . 6 ((x NC x ( NC ∩ ⋃1( ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c) “ Fin ))) ↔ (¬ x NC (x NC ( Spacx) Fin )))
3312, 32bitri 240 . . . . 5 (x ( ∼ NC ∪ ( NC ∩ ⋃1( ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c) “ Fin ))) ↔ (¬ x NC (x NC ( Spacx) Fin )))
34 andir 838 . . . . 5 (((¬ x NC x NC ) ( Spacx) Fin ) ↔ ((¬ x NC ( Spacx) Fin ) (x NC ( Spacx) Fin )))
3511, 33, 343bitr4i 268 . . . 4 (x ( ∼ NC ∪ ( NC ∩ ⋃1( ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c) “ Fin ))) ↔ ((¬ x NC x NC ) ( Spacx) Fin ))
361, 35mpbiran 884 . . 3 (x ( ∼ NC ∪ ( NC ∩ ⋃1( ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c) “ Fin ))) ↔ ( Spacx) Fin )
3736abbi2i 2464 . 2 ( ∼ NC ∪ ( NC ∩ ⋃1( ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c) “ Fin ))) = {x ( Spacx) Fin }
38 ncsex 6111 . . . 4 NC V
3938complex 4104 . . 3 NC V
40 ssetex 4744 . . . . . . . . . 10 S V
4140ins3ex 5798 . . . . . . . . 9 Ins3 S V
4240complex 4104 . . . . . . . . . . . . . 14 S V
4342cnvex 5102 . . . . . . . . . . . . 13 S V
4440cnvex 5102 . . . . . . . . . . . . . 14 S V
4520imageex 5801 . . . . . . . . . . . . . . . 16 Image{p, q (p NC q NC q = (2cc p))} V
4640, 45coex 4750 . . . . . . . . . . . . . . 15 ( S Image{p, q (p NC q NC q = (2cc p))}) V
4746fixex 5789 . . . . . . . . . . . . . 14 Fix ( S Image{p, q (p NC q NC q = (2cc p))}) V
4844, 47resex 5117 . . . . . . . . . . . . 13 ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})) V
4943, 48txpex 5785 . . . . . . . . . . . 12 ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))}))) V
5049rnex 5107 . . . . . . . . . . 11 ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))}))) V
5150complex 4104 . . . . . . . . . 10 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))}))) V
5251ins2ex 5797 . . . . . . . . 9 Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))}))) V
5341, 52symdifex 4108 . . . . . . . 8 ( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) V
54 1cex 4142 . . . . . . . 8 1c V
5553, 54imaex 4747 . . . . . . 7 (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c) V
5655complex 4104 . . . . . 6 ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c) V
57 finex 4397 . . . . . 6 Fin V
5856, 57imaex 4747 . . . . 5 ( ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c) “ Fin ) V
5958uni1ex 4293 . . . 4 1( ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c) “ Fin ) V
6038, 59inex 4105 . . 3 ( NC ∩ ⋃1( ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c) “ Fin )) V
6139, 60unex 4106 . 2 ( ∼ NC ∪ ( NC ∩ ⋃1( ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{p, q (p NC q NC q = (2cc p))})))) “ 1c) “ Fin ))) V
6237, 61eqeltrri 2424 1 {x ( Spacx) Fin } V
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 357   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  Vcvv 2859   ∼ ccompl 3205   ∪ cun 3207   ∩ cin 3208   ⊕ csymdif 3209  ∅c0 3550  {csn 3737  ⋃1cuni1 4133  1cc1c 4134   Fin cfin 4376  ⟨cop 4561  {copab 4622   class class class wbr 4639   S csset 4719   ∘ ccom 4721   “ cima 4722  ◡ccnv 4771  dom cdm 4772  ran crn 4773   ↾ cres 4774   Fn wfn 4776   ‘cfv 4781  (class class class)co 5525   ⊗ ctxp 5735   Fix cfix 5739   Ins2 cins2 5749   Ins3 cins3 5751  Imagecimage 5753   Clos1 cclos1 5872   NC cncs 6088  2cc2c 6094   ↑c cce 6096   Spac cspac 6273 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-fix 5740  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-pw1fn 5766  df-fullfun 5768  df-clos1 5873  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-map 6001  df-en 6029  df-ncs 6098  df-nc 6101  df-2c 6104  df-ce 6106  df-spac 6274 This theorem is referenced by:  nchoicelem19  6307
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