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Theorem spacvallem1 6281
 Description: Lemma for spacval 6282. Set up stratification for the recursive relationship. (Contributed by SF, 6-Mar-2015.)
Assertion
Ref Expression
spacvallem1 {x, y (x NC y NC y = (2cc x))} V
Distinct variable group:   x,y

Proof of Theorem spacvallem1
Dummy variable t is distinct from all other variables.
StepHypRef Expression
1 opelxp 4811 . . . . 5 (x, y ( NC × NC ) ↔ (x NC y NC ))
2 opelco 4884 . . . . . . 7 (x, y ( FullFunc (2nd (1st “ {2c}))) ↔ t(x(2nd (1st “ {2c}))t t FullFunc y))
3 brcnv 4892 . . . . . . . . . 10 (x(2nd (1st “ {2c}))tt(2nd (1st “ {2c}))x)
4 brres 4949 . . . . . . . . . . 11 (t(2nd (1st “ {2c}))x ↔ (t2nd x t (1st “ {2c})))
5 ancom 437 . . . . . . . . . . 11 ((t2nd x t (1st “ {2c})) ↔ (t (1st “ {2c}) t2nd x))
6 eliniseg 5020 . . . . . . . . . . . 12 (t (1st “ {2c}) ↔ t1st 2c)
76anbi1i 676 . . . . . . . . . . 11 ((t (1st “ {2c}) t2nd x) ↔ (t1st 2c t2nd x))
84, 5, 73bitri 262 . . . . . . . . . 10 (t(2nd (1st “ {2c}))x ↔ (t1st 2c t2nd x))
9 2nc 6168 . . . . . . . . . . . 12 2c NC
109elexi 2868 . . . . . . . . . . 11 2c V
11 vex 2862 . . . . . . . . . . 11 x V
1210, 11op1st2nd 5790 . . . . . . . . . 10 ((t1st 2c t2nd x) ↔ t = 2c, x)
133, 8, 123bitri 262 . . . . . . . . 9 (x(2nd (1st “ {2c}))tt = 2c, x)
1413anbi1i 676 . . . . . . . 8 ((x(2nd (1st “ {2c}))t t FullFunc y) ↔ (t = 2c, x t FullFunc y))
1514exbii 1582 . . . . . . 7 (t(x(2nd (1st “ {2c}))t t FullFunc y) ↔ t(t = 2c, x t FullFunc y))
162, 15bitri 240 . . . . . 6 (x, y ( FullFunc (2nd (1st “ {2c}))) ↔ t(t = 2c, x t FullFunc y))
1710, 11opex 4588 . . . . . . 7 2c, x V
18 breq1 4642 . . . . . . 7 (t = 2c, x → (t FullFunc y2c, x FullFunc y))
1917, 18ceqsexv 2894 . . . . . 6 (t(t = 2c, x t FullFunc y) ↔ 2c, x FullFunc y)
2010, 11brfullfunop 5867 . . . . . . 7 (2c, x FullFunc y ↔ (2cc x) = y)
21 eqcom 2355 . . . . . . 7 ((2cc x) = yy = (2cc x))
2220, 21bitri 240 . . . . . 6 (2c, x FullFunc yy = (2cc x))
2316, 19, 223bitri 262 . . . . 5 (x, y ( FullFunc (2nd (1st “ {2c}))) ↔ y = (2cc x))
241, 23anbi12i 678 . . . 4 ((x, y ( NC × NC ) x, y ( FullFunc (2nd (1st “ {2c})))) ↔ ((x NC y NC ) y = (2cc x)))
25 elin 3219 . . . 4 (x, y (( NC × NC ) ∩ ( FullFunc (2nd (1st “ {2c})))) ↔ (x, y ( NC × NC ) x, y ( FullFunc (2nd (1st “ {2c})))))
26 df-3an 936 . . . 4 ((x NC y NC y = (2cc x)) ↔ ((x NC y NC ) y = (2cc x)))
2724, 25, 263bitr4i 268 . . 3 (x, y (( NC × NC ) ∩ ( FullFunc (2nd (1st “ {2c})))) ↔ (x NC y NC y = (2cc x)))
2827opabbi2i 4866 . 2 (( NC × NC ) ∩ ( FullFunc (2nd (1st “ {2c})))) = {x, y (x NC y NC y = (2cc x))}
29 ncsex 6111 . . . 4 NC V
3029, 29xpex 5115 . . 3 ( NC × NC ) V
31 ceex 6174 . . . . 5 c V
3231fullfunex 5860 . . . 4 FullFunc V
33 2ndex 5112 . . . . . 6 2nd V
34 1stex 4739 . . . . . . . 8 1st V
3534cnvex 5102 . . . . . . 7 1st V
36 snex 4111 . . . . . . 7 {2c} V
3735, 36imaex 4747 . . . . . 6 (1st “ {2c}) V
3833, 37resex 5117 . . . . 5 (2nd (1st “ {2c})) V
3938cnvex 5102 . . . 4 (2nd (1st “ {2c})) V
4032, 39coex 4750 . . 3 ( FullFunc (2nd (1st “ {2c}))) V
4130, 40inex 4105 . 2 (( NC × NC ) ∩ ( FullFunc (2nd (1st “ {2c})))) V
4228, 41eqeltrri 2424 1 {x, y (x NC y NC y = (2cc x))} V
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∩ cin 3208  {csn 3737  ⟨cop 4561  {copab 4622   class class class wbr 4639  1st c1st 4717   ∘ ccom 4721   “ cima 4722   × cxp 4770  ◡ccnv 4771   ↾ cres 4774  2nd c2nd 4783  (class class class)co 5525   FullFun cfullfun 5767   NC cncs 6088  2cc2c 6094   ↑c cce 6096 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-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-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 This theorem is referenced by:  spacval  6282  fnspac  6283  spacssnc  6284  spacind  6287  nchoicelem3  6291  nchoicelem11  6299  nchoicelem16  6304  nchoicelem18  6306
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