Theorem spacvallem1 6282

Description: Lemma for spacval 6283. Set up stratification for the recursive relationship. (Contributed by SF, 6-Mar-2015.)

Assertion

Ref	Expression
spacvallem1	⊢ {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))} ∈ V

Distinct variable group:   x,y

Proof of Theorem spacvallem1

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opelxp 4812	. . . . 5 ⊢ (⟨x, y⟩ ∈ ( NC × NC ) ↔ (x ∈ NC ∧ y ∈ NC ))
2		opelco 4885	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ ( FullFun ↑c ∘ ◡(2nd ↾ (◡1st “ {2c}))) ↔ ∃t(x◡(2nd ↾ (◡1st “ {2c}))t ∧ t FullFun ↑c y))
3		brcnv 4893	. . . . . . . . . 10 ⊢ (x◡(2nd ↾ (◡1st “ {2c}))t ↔ t(2nd ↾ (◡1st “ {2c}))x)
4		brres 4950	. . . . . . . . . . 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 5021	. . . . . . . . . . . 12 ⊢ (t ∈ (◡1st “ {2c}) ↔ t1st 2c)
7	6	anbi1i 676	. . . . . . . . . . 11 ⊢ ((t ∈ (◡1st “ {2c}) ∧ t2nd x) ↔ (t1st 2c ∧ t2nd x))
8	4, 5, 7	3bitri 262	. . . . . . . . . 10 ⊢ (t(2nd ↾ (◡1st “ {2c}))x ↔ (t1st 2c ∧ t2nd x))
9		2nc 6169	. . . . . . . . . . . 12 ⊢ 2c ∈ NC
10	9	elexi 2869	. . . . . . . . . . 11 ⊢ 2c ∈ V
11		vex 2863	. . . . . . . . . . 11 ⊢ x ∈ V
12	10, 11	op1st2nd 5791	. . . . . . . . . 10 ⊢ ((t1st 2c ∧ t2nd x) ↔ t = ⟨2c, x⟩)
13	3, 8, 12	3bitri 262	. . . . . . . . 9 ⊢ (x◡(2nd ↾ (◡1st “ {2c}))t ↔ t = ⟨2c, x⟩)
14	13	anbi1i 676	. . . . . . . 8 ⊢ ((x◡(2nd ↾ (◡1st “ {2c}))t ∧ t FullFun ↑c y) ↔ (t = ⟨2c, x⟩ ∧ t FullFun ↑c y))
15	14	exbii 1582	. . . . . . 7 ⊢ (∃t(x◡(2nd ↾ (◡1st “ {2c}))t ∧ t FullFun ↑c y) ↔ ∃t(t = ⟨2c, x⟩ ∧ t FullFun ↑c y))
16	2, 15	bitri 240	. . . . . 6 ⊢ (⟨x, y⟩ ∈ ( FullFun ↑c ∘ ◡(2nd ↾ (◡1st “ {2c}))) ↔ ∃t(t = ⟨2c, x⟩ ∧ t FullFun ↑c y))
17	10, 11	opex 4589	. . . . . . 7 ⊢ ⟨2c, x⟩ ∈ V
18		breq1 4643	. . . . . . 7 ⊢ (t = ⟨2c, x⟩ → (t FullFun ↑c y ↔ ⟨2c, x⟩ FullFun ↑c y))
19	17, 18	ceqsexv 2895	. . . . . 6 ⊢ (∃t(t = ⟨2c, x⟩ ∧ t FullFun ↑c y) ↔ ⟨2c, x⟩ FullFun ↑c y)
20	10, 11	brfullfunop 5868	. . . . . . 7 ⊢ (⟨2c, x⟩ FullFun ↑c y ↔ (2c ↑c x) = y)
21		eqcom 2355	. . . . . . 7 ⊢ ((2c ↑c x) = y ↔ y = (2c ↑c x))
22	20, 21	bitri 240	. . . . . 6 ⊢ (⟨2c, x⟩ FullFun ↑c y ↔ y = (2c ↑c x))
23	16, 19, 22	3bitri 262	. . . . 5 ⊢ (⟨x, y⟩ ∈ ( FullFun ↑c ∘ ◡(2nd ↾ (◡1st “ {2c}))) ↔ y = (2c ↑c x))
24	1, 23	anbi12i 678	. . . 4 ⊢ ((⟨x, y⟩ ∈ ( NC × NC ) ∧ ⟨x, y⟩ ∈ ( FullFun ↑c ∘ ◡(2nd ↾ (◡1st “ {2c})))) ↔ ((x ∈ NC ∧ y ∈ NC ) ∧ y = (2c ↑c x)))
25		elin 3220	. . . 4 ⊢ (⟨x, y⟩ ∈ (( NC × NC ) ∩ ( FullFun ↑c ∘ ◡(2nd ↾ (◡1st “ {2c})))) ↔ (⟨x, y⟩ ∈ ( NC × NC ) ∧ ⟨x, y⟩ ∈ ( FullFun ↑c ∘ ◡(2nd ↾ (◡1st “ {2c})))))
26		df-3an 936	. . . 4 ⊢ ((x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x)) ↔ ((x ∈ NC ∧ y ∈ NC ) ∧ y = (2c ↑c x)))
27	24, 25, 26	3bitr4i 268	. . 3 ⊢ (⟨x, y⟩ ∈ (( NC × NC ) ∩ ( FullFun ↑c ∘ ◡(2nd ↾ (◡1st “ {2c})))) ↔ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x)))
28	27	opabbi2i 4867	. 2 ⊢ (( NC × NC ) ∩ ( FullFun ↑c ∘ ◡(2nd ↾ (◡1st “ {2c})))) = {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))}
29		ncsex 6112	. . . 4 ⊢ NC ∈ V
30	29, 29	xpex 5116	. . 3 ⊢ ( NC × NC ) ∈ V
31		ceex 6175	. . . . 5 ⊢ ↑c ∈ V
32	31	fullfunex 5861	. . . 4 ⊢ FullFun ↑c ∈ V
33		2ndex 5113	. . . . . 6 ⊢ 2nd ∈ V
34		1stex 4740	. . . . . . . 8 ⊢ 1st ∈ V
35	34	cnvex 5103	. . . . . . 7 ⊢ ◡1st ∈ V
36		snex 4112	. . . . . . 7 ⊢ {2c} ∈ V
37	35, 36	imaex 4748	. . . . . 6 ⊢ (◡1st “ {2c}) ∈ V
38	33, 37	resex 5118	. . . . 5 ⊢ (2nd ↾ (◡1st “ {2c})) ∈ V
39	38	cnvex 5103	. . . 4 ⊢ ◡(2nd ↾ (◡1st “ {2c})) ∈ V
40	32, 39	coex 4751	. . 3 ⊢ ( FullFun ↑c ∘ ◡(2nd ↾ (◡1st “ {2c}))) ∈ V
41	30, 40	inex 4106	. 2 ⊢ (( NC × NC ) ∩ ( FullFun ↑c ∘ ◡(2nd ↾ (◡1st “ {2c})))) ∈ V
42	28, 41	eqeltrri 2424	1 ⊢ {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))} ∈ V

Syntax hints:   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∩ cin 3209  {csn 3738  ⟨cop 4562  {copab 4623   class class class wbr 4640  1^st c1st 4718   ∘ ccom 4722   “ cima 4723   × cxp 4771  ^◡ccnv 4772   ↾ cres 4775  2^nd c2nd 4784  (class class class)co 5526   FullFun cfullfun 5768   NC cncs 6089  2_cc2c 6095   ↑_c cce 6097

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-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-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

This theorem is referenced by:  spacval  6283  fnspac  6284  spacssnc  6285  spacind  6288  nchoicelem3  6292  nchoicelem11  6300  nchoicelem16  6305  nchoicelem18  6307