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 ∈ V
clos1ex.2	⊢ R ∈ V

Assertion

Ref	Expression
clos1ex	⊢ Clos1 (S, R) ∈ 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 ⊆ a ∧ (R “ a) ⊆ a)}
2		elin 3220	. . . . . 6 ⊢ (a ∈ (( S “ {S}) ∩ Fix ( S ∘ ImageR)) ↔ (a ∈ ( S “ {S}) ∧ a ∈ Fix ( S ∘ ImageR)))
3		elimasn 5020	. . . . . . . 8 ⊢ (a ∈ ( S “ {S}) ↔ ⟨S, a⟩ ∈ S )
4		df-br 4641	. . . . . . . 8 ⊢ (S S a ↔ ⟨S, a⟩ ∈ S )
5		clos1ex.1	. . . . . . . . 9 ⊢ S ∈ V
6		vex 2863	. . . . . . . . 9 ⊢ a ∈ V
7	5, 6	brsset 4759	. . . . . . . 8 ⊢ (S S a ↔ S ⊆ a)
8	3, 4, 7	3bitr2i 264	. . . . . . 7 ⊢ (a ∈ ( S “ {S}) ↔ S ⊆ a)
9		elfix 5788	. . . . . . . 8 ⊢ (a ∈ Fix ( S ∘ ImageR) ↔ a( S ∘ ImageR)a)
10		brco 4884	. . . . . . . . 9 ⊢ (a( S ∘ ImageR)a ↔ ∃b(aImageRb ∧ b S a))
11		vex 2863	. . . . . . . . . . . . 13 ⊢ b ∈ 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 ⊢ (∃b(aImageRb ∧ b S a) ↔ ∃b(b = (R “ a) ∧ b S a))
15		clos1ex.2	. . . . . . . . . . . 12 ⊢ R ∈ V
16	15, 6	imaex 4748	. . . . . . . . . . 11 ⊢ (R “ a) ∈ 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) ⊆ a)
19	17, 18	syl6bb 252	. . . . . . . . . . 11 ⊢ (b = (R “ a) → (b S a ↔ (R “ a) ⊆ a))
20	16, 19	ceqsexv 2895	. . . . . . . . . 10 ⊢ (∃b(b = (R “ a) ∧ b S a) ↔ (R “ a) ⊆ a)
21	14, 20	bitri 240	. . . . . . . . 9 ⊢ (∃b(aImageRb ∧ b S a) ↔ (R “ a) ⊆ a)
22	10, 21	bitri 240	. . . . . . . 8 ⊢ (a( S ∘ ImageR)a ↔ (R “ a) ⊆ a)
23	9, 22	bitri 240	. . . . . . 7 ⊢ (a ∈ Fix ( S ∘ ImageR) ↔ (R “ a) ⊆ a)
24	8, 23	anbi12i 678	. . . . . 6 ⊢ ((a ∈ ( S “ {S}) ∧ a ∈ Fix ( S ∘ ImageR)) ↔ (S ⊆ a ∧ (R “ a) ⊆ a))
25	2, 24	bitri 240	. . . . 5 ⊢ (a ∈ (( S “ {S}) ∩ Fix ( S ∘ ImageR)) ↔ (S ⊆ a ∧ (R “ a) ⊆ a))
26	25	abbi2i 2465	. . . 4 ⊢ (( S “ {S}) ∩ Fix ( S ∘ ImageR)) = {a ∣ (S ⊆ a ∧ (R “ a) ⊆ a)}
27		ssetex 4745	. . . . . 6 ⊢ S ∈ V
28		snex 4112	. . . . . 6 ⊢ {S} ∈ V
29	27, 28	imaex 4748	. . . . 5 ⊢ ( S “ {S}) ∈ V
30	15	imageex 5802	. . . . . . 7 ⊢ ImageR ∈ V
31	27, 30	coex 4751	. . . . . 6 ⊢ ( S ∘ ImageR) ∈ V
32	31	fixex 5790	. . . . 5 ⊢ Fix ( S ∘ ImageR) ∈ V
33	29, 32	inex 4106	. . . 4 ⊢ (( S “ {S}) ∩ Fix ( S ∘ ImageR)) ∈ V
34	26, 33	eqeltrri 2424	. . 3 ⊢ {a ∣ (S ⊆ a ∧ (R “ a) ⊆ a)} ∈ V
35	34	intex 4321	. 2 ⊢ ∩{a ∣ (S ⊆ a ∧ (R “ a) ⊆ a)} ∈ V
36	1, 35	eqeltri 2423	1 ⊢ Clos1 (S, R) ∈ V

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2860   ∩ cin 3209   ⊆ wss 3258  {csn 3738  ∩cint 3927  ⟨cop 4562   class class class wbr 4640   S csset 4720   ∘ ccom 4722   “ cima 4723   Fix cfix 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