Theorem spacis 6289

Description: Induction scheme for the special set generator. (Contributed by SF, 13-Mar-2015.)

Hypotheses

Ref	Expression
spacis.1	|- {x | ph} e. _V
spacis.2	|- (x = M -> (ph <-> ps))
spacis.3	|- (x = y -> (ph <-> ch))
spacis.4	|- (x = (2c ↑c y) -> (ph <-> th))
spacis.5	|- (x = N -> (ph <-> ta))
spacis.6	|- (M e. NC -> ps)
spacis.7	|- (((M e. NC /\ y e. ( Spac ` M)) /\ ((y ↑c 0c) e. NC /\ ch)) -> th)

Assertion

Ref	Expression
spacis	|- ((M e. NC /\ N e. ( Spac ` M)) -> ta)

Distinct variable groups:   ch,x   x,M,y   x,N   ph,y   ps,x   ta,x   th,x   x,y

Allowed substitution hints:   ph(x)   ps(y)   ch(y)   th(y)   ta(y)   N(y)

Proof of Theorem spacis

Step	Hyp	Ref	Expression
1		id 19	. . . 4 |- (M e. NC -> M e. NC )
2		spacis.1	. . . . 5 |- {x | ph} e. _V
3	2	a1i 10	. . . 4 |- (M e. NC -> {x | ph} e. _V)
4		spacis.6	. . . . 5 |- (M e. NC -> ps)
5		spacis.2	. . . . . 6 |- (x = M -> (ph <-> ps))
6	5	elabg 2987	. . . . 5 |- (M e. NC -> (M e. {x | ph} <-> ps))
7	4, 6	mpbird 223	. . . 4 |- (M e. NC -> M e. {x | ph})
8		ancom 437	. . . . . . 7 |- ((y e. {x | ph} /\ (y ↑c 0c) e. NC ) <-> ((y ↑c 0c) e. NC /\ y e. {x | ph}))
9		vex 2863	. . . . . . . . 9 |- y e. _V
10		spacis.3	. . . . . . . . 9 |- (x = y -> (ph <-> ch))
11	9, 10	elab 2986	. . . . . . . 8 |- (y e. {x | ph} <-> ch)
12	11	anbi2i 675	. . . . . . 7 |- (((y ↑c 0c) e. NC /\ y e. {x | ph}) <-> ((y ↑c 0c) e. NC /\ ch))
13	8, 12	bitri 240	. . . . . 6 |- ((y e. {x | ph} /\ (y ↑c 0c) e. NC ) <-> ((y ↑c 0c) e. NC /\ ch))
14		spacis.7	. . . . . . . 8 |- (((M e. NC /\ y e. ( Spac ` M)) /\ ((y ↑c 0c) e. NC /\ ch)) -> th)
15		ovex 5552	. . . . . . . . 9 |- (2c ↑c y) e. _V
16		spacis.4	. . . . . . . . 9 |- (x = (2c ↑c y) -> (ph <-> th))
17	15, 16	elab 2986	. . . . . . . 8 |- ((2c ↑c y) e. {x | ph} <-> th)
18	14, 17	sylibr 203	. . . . . . 7 |- (((M e. NC /\ y e. ( Spac ` M)) /\ ((y ↑c 0c) e. NC /\ ch)) -> (2c ↑c y) e. {x | ph})
19	18	ex 423	. . . . . 6 |- ((M e. NC /\ y e. ( Spac ` M)) -> (((y ↑c 0c) e. NC /\ ch) -> (2c ↑c y) e. {x | ph}))
20	13, 19	syl5bi 208	. . . . 5 |- ((M e. NC /\ y e. ( Spac ` M)) -> ((y e. {x | ph} /\ (y ↑c 0c) e. NC ) -> (2c ↑c y) e. {x | ph}))
21	20	ralrimiva 2698	. . . 4 |- (M e. NC -> A.y e. ( Spac ` M)((y e. {x | ph} /\ (y ↑c 0c) e. NC ) -> (2c ↑c y) e. {x | ph}))
22		spacind 6288	. . . 4 |- (((M e. NC /\ {x | ph} e. _V) /\ (M e. {x | ph} /\ A.y e. ( Spac ` M)((y e. {x | ph} /\ (y ↑c 0c) e. NC ) -> (2c ↑c y) e. {x | ph}))) -> ( Spac ` M) C_ {x | ph})
23	1, 3, 7, 21, 22	syl22anc 1183	. . 3 |- (M e. NC -> ( Spac ` M) C_ {x | ph})
24	23	sselda 3274	. 2 |- ((M e. NC /\ N e. ( Spac ` M)) -> N e. {x | ph})
25		spacis.5	. . . 4 |- (x = N -> (ph <-> ta))
26	25	elabg 2987	. . 3 |- (N e. ( Spac ` M) -> (N e. {x | ph} <-> ta))
27	26	adantl 452	. 2 |- ((M e. NC /\ N e. ( Spac ` M)) -> (N e. {x | ph} <-> ta))
28	24, 27	mpbid 201	1 |- ((M e. NC /\ N e. ( Spac ` M)) -> ta)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  {cab 2339  A.wral 2615  _Vcvv 2860   C_ wss 3258  0_cc0c 4375  ` cfv 4782  (class class class)co 5526   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-compose 5749  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:  nchoicelem4  6293