Theorem clos1ex 5876

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 5873	. 2 |- Clos1 (S, R) = |^|{a | (S C_ a /\ (R"a) C_ a)}
2		elin 3219	. . . . . 6 |- (a e. (( S "{S}) i^i Fix( S o. ImageR)) <-> (a e. ( S "{S}) /\ a e. Fix( S o. ImageR)))
3		elimasn 5019	. . . . . . . 8 |- (a e. ( S "{S}) <-> <.S, a>. e. S )
4		df-br 4640	. . . . . . . 8 |- (S S a <-> <.S, a>. e. S )
5		clos1ex.1	. . . . . . . . 9 |- S e. _V
6		vex 2862	. . . . . . . . 9 |- a e. _V
7	5, 6	brsset 4758	. . . . . . . 8 |- (S S a <-> S C_ a)
8	3, 4, 7	3bitr2i 264	. . . . . . 7 |- (a e. ( S "{S}) <-> S C_ a)
9		elfix 5787	. . . . . . . 8 |- (a e. Fix( S o. ImageR) <-> a( S o. ImageR)a)
10		brco 4883	. . . . . . . . 9 |- (a( S o. ImageR)a <-> E.b(aImageRb /\ b S a))
11		vex 2862	. . . . . . . . . . . . 13 |- b e. _V
12	6, 11	brimage 5793	. . . . . . . . . . . 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 4747	. . . . . . . . . . 11 |- (R"a) e. _V
17		breq1 4642	. . . . . . . . . . . 12 |- (b = (R"a) -> (b S a <-> (R"a) S a))
18	16, 6	brsset 4758	. . . . . . . . . . . 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 2894	. . . . . . . . . 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 2464	. . . 4 |- (( S "{S}) i^i Fix( S o. ImageR)) = {a | (S C_ a /\ (R"a) C_ a)}
27		ssetex 4744	. . . . . 6 |- S e. _V
28		snex 4111	. . . . . 6 |- {S} e. _V
29	27, 28	imaex 4747	. . . . 5 |- ( S "{S}) e. _V
30	15	imageex 5801	. . . . . . 7 |- ImageR e. _V
31	27, 30	coex 4750	. . . . . 6 |- ( S o. ImageR) e. _V
32	31	fixex 5789	. . . . 5 |- Fix( S o. ImageR) e. _V
33	29, 32	inex 4105	. . . 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 4320	. 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 2859   i^i cin 3208   C_ wss 3257  {csn 3737  |^|cint 3926  <.cop 4561   class class class wbr 4639   S csset 4719   o. ccom 4721  "cima 4722  Fixcfix 5739  Imagecimage 5753   Clos1 cclos1 5872

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-2nd 4797  df-txp 5736  df-fix 5740  df-ins2 5750  df-ins3 5752  df-image 5754  df-clos1 5873

This theorem is referenced by:  clos1exg  5877  clos1induct  5880  clos1basesuc  5882  sbthlem1  6203  spacval  6282  fnspac  6283  nchoicelem11  6299  nchoicelem16  6304