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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 Spac Fin

Proof of Theorem nchoicelem18
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pm2.1 406 . . . 4 NC NC
2 fnspac 6283 . . . . . . . . . . 11 Spac NC
3 fndm 5219 . . . . . . . . . . 11 Spac NC Spac NC
42, 3ax-mp 8 . . . . . . . . . 10 Spac NC
54eleq2i 2417 . . . . . . . . 9 Spac NC
6 ndmfv 5389 . . . . . . . . 9 Spac Spac
75, 6sylnbir 298 . . . . . . . 8 NC Spac
8 0fin 4422 . . . . . . . 8 Fin
97, 8syl6eqel 2441 . . . . . . 7 NC Spac Fin
109pm4.71i 613 . . . . . 6 NC NC Spac Fin
1110orbi1i 506 . . . . 5 NC NC Spac Fin NC Spac Fin NC Spac Fin
12 elun 3220 . . . . . 6 NC NC 1 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Fin NC NC 1 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Fin
13 vex 2862 . . . . . . . 8
1413elcompl 3225 . . . . . . 7 NC NC
15 elin 3219 . . . . . . . 8 NC 1 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Fin NC 1 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Fin
16 spacval 6282 . . . . . . . . . . 11 NC Spac Clos1 NC NC 2cc
1716eleq1d 2419 . . . . . . . . . 10 NC Spac Fin Clos1 NC NC 2cc Fin
1813eluni1 4173 . . . . . . . . . . 11 1 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Fin Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Fin
19 df-br 4632 . . . . . . . . . . . . . 14 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c
20 spacvallem1 6281 . . . . . . . . . . . . . . 15 NC NC 2cc
21 snex 4111 . . . . . . . . . . . . . . 15
2220, 21nchoicelem10 6298 . . . . . . . . . . . . . 14 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Clos1 NC NC 2cc
2319, 22bitri 240 . . . . . . . . . . . . 13 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Clos1 NC NC 2cc
2423rexbii 2639 . . . . . . . . . . . 12 Fin Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Fin Clos1 NC NC 2cc
25 elima 4746 . . . . . . . . . . . 12 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Fin Fin Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c
26 risset 2661 . . . . . . . . . . . 12 Clos1 NC NC 2cc Fin Fin Clos1 NC NC 2cc
2724, 25, 263bitr4i 268 . . . . . . . . . . 11 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Fin Clos1 NC NC 2cc Fin
2818, 27bitri 240 . . . . . . . . . 10 1 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Fin Clos1 NC NC 2cc Fin
2917, 28syl6rbbr 255 . . . . . . . . 9 NC 1 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Fin Spac Fin
3029pm5.32i 618 . . . . . . . 8 NC 1 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Fin NC Spac Fin
3115, 30bitri 240 . . . . . . 7 NC 1 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Fin NC Spac Fin
3214, 31orbi12i 507 . . . . . 6 NC NC 1 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Fin NC NC Spac Fin
3312, 32bitri 240 . . . . 5 NC NC 1 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Fin NC NC Spac Fin
34 andir 838 . . . . 5 NC NC Spac Fin NC Spac Fin NC Spac Fin
3511, 33, 343bitr4i 268 . . . 4 NC NC 1 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Fin NC NC Spac Fin
361, 35mpbiran 884 . . 3 NC NC 1 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Fin Spac Fin
3736abbi2i 2464 . 2 NC NC 1 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Fin Spac Fin
38 ncsex 6131 . . . 4 NC
3938complex 4104 . . 3 NC
40 ssetex 4736 . . . . . . . . . 10 SSet
4140ins3ex 5827 . . . . . . . . 9 Ins3 SSet
4240complex 4104 . . . . . . . . . . . . . 14 SSet
4342cnvex 5135 . . . . . . . . . . . . 13 SSet
4440cnvex 5135 . . . . . . . . . . . . . 14 SSet
4520imageex 5830 . . . . . . . . . . . . . . . 16 Image NC NC 2cc
4640, 45coex 4742 . . . . . . . . . . . . . . 15 SSet Image NC NC 2cc
4746fixex 5818 . . . . . . . . . . . . . 14 SSet Image NC NC 2cc
4844, 47resex 5150 . . . . . . . . . . . . 13 SSet SSet Image NC NC 2cc
4943, 48txpex 5813 . . . . . . . . . . . 12 SSet SSet SSet Image NC NC 2cc
5049rnex 5140 . . . . . . . . . . 11 SSet SSet SSet Image NC NC 2cc
5150complex 4104 . . . . . . . . . 10 SSet SSet SSet Image NC NC 2cc
5251ins2ex 5826 . . . . . . . . 9 Ins2 SSet SSet SSet Image NC NC 2cc
5341, 52symdifex 4108 . . . . . . . 8 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc
54 1cex 4142 . . . . . . . 8 1c
5553, 54imaex 4739 . . . . . . 7 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c
5655complex 4104 . . . . . 6 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c
57 finex 4397 . . . . . 6 Fin
5856, 57imaex 4739 . . . . 5 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Fin
5958uni1ex 4293 . . . 4 1 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Fin
6038, 59inex 4105 . . 3 NC 1 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Fin
6139, 60unex 4106 . 2 NC NC 1 Ins3 SSet Ins2 SSet SSet SSet Image NC NC 2cc 1c Fin
6237, 61eqeltrri 2424 1 Spac Fin
Colors of variables: wff set class
Syntax hints:   wn 3   wo 357   wa 358   w3a 934   wceq 1642   wcel 1710  cab 2339  wrex 2615  cvv 2859   ∼ ccompl 3205   cun 3207   cin 3208   csymdif 3209  c0 3550  csn 3737  ⋃1cuni1 4133  1cc1c 4134   Fin cfin 4376  cop 4561  copab 4614   class class class wbr 4631   SSet csset 4711   ccom 4713  cima 4714  ccnv 4763   cdm 4764   crn 4765   cres 4766   wfn 4769  cfv 4774  (class class class)co 5564   ctxp 5771  cfix 5773  Imagecimage 5774   Ins2 cins2 5778   Ins3 cins3 5779   Clos1 cclos1 5894   NC cncs 6108  2cc2c 6114   ↑c cce 6116   Spac cspac 6273
This theorem is referenced by:  nchoicelem19  6307
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-4 2135  ax-5o 2136  ax-6o 2137  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 4439  df-ltfin 4440  df-ncfin 4441  df-tfin 4442  df-evenfin 4443  df-oddfin 4444  df-sfin 4445  df-spfin 4446  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4615  df-br 4632  df-1st 4715  df-swap 4716  df-sset 4717  df-co 4718  df-ima 4719  df-si 4720  df-id 4759  df-xp 4777  df-rel 4778  df-cnv 4779  df-rn 4780  df-dm 4781  df-res 4782  df-fun 4783  df-fn 4784  df-f 4785  df-f1 4786  df-fo 4787  df-f1o 4788  df-fv 4789  df-2nd 4791  df-ov 5566  df-oprab 5567  df-mpt 5693  df-mpt2 5694  df-txp 5787  df-fix 5789  df-ins2 5793  df-ins3 5794  df-image 5795  df-ins4 5796  df-si3 5797  df-funs 5798  df-fns 5799  df-pw1fn 5801  df-fullfun 5802  df-clos1 5895  df-trans 5919  df-sym 5928  df-er 5929  df-ec 5967  df-qs 5971  df-map 6021  df-en 6049  df-ncs 6118  df-nc 6121  df-2c 6124  df-ce 6126  df-spac 6274
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