Theorem clos1ex 5877

Description: The closure of a set under a set is a set. (Contributed by SF, 11-Feb-2015.)

Hypotheses

Ref	Expression
clos1ex.1	|- S e. _V
clos1ex.2	|- R e. _V

Assertion

Ref	Expression
clos1ex	|- Clos1 (S, R) e. _V

Proof of Theorem clos1ex

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clos1 5874	. 2 |- Clos1 (S, R) = |^|{a | (S C_ a /\ (R"a) C_ a)}
2		elin 3220	. . . . . 6 |- (a e. (( S "{S}) i^i Fix( S o. ImageR)) <-> (a e. ( S "{S}) /\ a e. Fix( S o. ImageR)))
3		elimasn 5020	. . . . . . . 8 |- (a e. ( S "{S}) <-> <.S, a>. e. S )
4		df-br 4641	. . . . . . . 8 |- (S S a <-> <.S, a>. e. S )
5		clos1ex.1	. . . . . . . . 9 |- S e. _V
6		vex 2863	. . . . . . . . 9 |- a e. _V
7	5, 6	brsset 4759	. . . . . . . 8 |- (S S a <-> S C_ a)
8	3, 4, 7	3bitr2i 264	. . . . . . 7 |- (a e. ( S "{S}) <-> S C_ a)
9		elfix 5788	. . . . . . . 8 |- (a e. Fix( S o. ImageR) <-> a( S o. ImageR)a)
10		brco 4884	. . . . . . . . 9 |- (a( S o. ImageR)a <-> E.b(aImageRb /\ b S a))
11		vex 2863	. . . . . . . . . . . . 13 |- b e. _V
12	6, 11	brimage 5794	. . . . . . . . . . . 12 |- (aImageRb <-> b = (R"a))
13	12	anbi1i 676	. . . . . . . . . . 11 |- ((aImageRb /\ b S a) <-> (b = (R"a) /\ b S a))
14	13	exbii 1582	. . . . . . . . . 10 |- (E.b(aImageRb /\ b S a) <-> E.b(b = (R"a) /\ b S a))
15		clos1ex.2	. . . . . . . . . . . 12 |- R e. _V
16	15, 6	imaex 4748	. . . . . . . . . . 11 |- (R"a) e. _V
17		breq1 4643	. . . . . . . . . . . 12 |- (b = (R"a) -> (b S a <-> (R"a) S a))
18	16, 6	brsset 4759	. . . . . . . . . . . 12 |- ((R"a) S a <-> (R"a) C_ a)
19	17, 18	syl6bb 252	. . . . . . . . . . 11 |- (b = (R"a) -> (b S a <-> (R"a) C_ a))
20	16, 19	ceqsexv 2895	. . . . . . . . . 10 |- (E.b(b = (R"a) /\ b S a) <-> (R"a) C_ a)
21	14, 20	bitri 240	. . . . . . . . 9 |- (E.b(aImageRb /\ b S a) <-> (R"a) C_ a)
22	10, 21	bitri 240	. . . . . . . 8 |- (a( S o. ImageR)a <-> (R"a) C_ a)
23	9, 22	bitri 240	. . . . . . 7 |- (a e. Fix( S o. ImageR) <-> (R"a) C_ a)
24	8, 23	anbi12i 678	. . . . . 6 |- ((a e. ( S "{S}) /\ a e. Fix( S o. ImageR)) <-> (S C_ a /\ (R"a) C_ a))
25	2, 24	bitri 240	. . . . 5 |- (a e. (( S "{S}) i^i Fix( S o. ImageR)) <-> (S C_ a /\ (R"a) C_ a))
26	25	abbi2i 2465	. . . 4 |- (( S "{S}) i^i Fix( S o. ImageR)) = {a | (S C_ a /\ (R"a) C_ a)}
27		ssetex 4745	. . . . . 6 |- S e. _V
28		snex 4112	. . . . . 6 |- {S} e. _V
29	27, 28	imaex 4748	. . . . 5 |- ( S "{S}) e. _V
30	15	imageex 5802	. . . . . . 7 |- ImageR e. _V
31	27, 30	coex 4751	. . . . . 6 |- ( S o. ImageR) e. _V
32	31	fixex 5790	. . . . 5 |- Fix( S o. ImageR) e. _V
33	29, 32	inex 4106	. . . 4 |- (( S "{S}) i^i Fix( S o. ImageR)) e. _V
34	26, 33	eqeltrri 2424	. . 3 |- {a | (S C_ a /\ (R"a) C_ a)} e. _V
35	34	intex 4321	. 2 |- |^|{a | (S C_ a /\ (R"a) C_ a)} e. _V
36	1, 35	eqeltri 2423	1 |- Clos1 (S, R) e. _V

Syntax hints:   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339  _Vcvv 2860   i^i cin 3209   C_ wss 3258  {csn 3738  |^|cint 3927  <.cop 4562   class class class wbr 4640   S csset 4720   o. ccom 4722  "cima 4723  Fixcfix 5740  Imagecimage 5754   Clos1 cclos1 5873

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-2nd 4798  df-txp 5737  df-fix 5741  df-ins2 5751  df-ins3 5753  df-image 5755  df-clos1 5874

This theorem is referenced by:  clos1exg  5878  clos1induct  5881  clos1basesuc  5883  sbthlem1  6204  spacval  6283  fnspac  6284  nchoicelem11  6300  nchoicelem16  6305