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 ∈ 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 5873	. 2 ⊢ Clos1 (S, R) = ∩{a ∣ (S ⊆ a ∧ (R “ a) ⊆ a)}
2		elin 3219	. . . . . 6 ⊢ (a ∈ (( S “ {S}) ∩ Fix ( S ∘ ImageR)) ↔ (a ∈ ( S “ {S}) ∧ a ∈ Fix ( S ∘ ImageR)))
3		elimasn 5019	. . . . . . . 8 ⊢ (a ∈ ( S “ {S}) ↔ ⟨S, a⟩ ∈ S )
4		df-br 4640	. . . . . . . 8 ⊢ (S S a ↔ ⟨S, a⟩ ∈ S )
5		clos1ex.1	. . . . . . . . 9 ⊢ S ∈ V
6		vex 2862	. . . . . . . . 9 ⊢ a ∈ V
7	5, 6	brsset 4758	. . . . . . . 8 ⊢ (S S a ↔ S ⊆ a)
8	3, 4, 7	3bitr2i 264	. . . . . . 7 ⊢ (a ∈ ( S “ {S}) ↔ S ⊆ a)
9		elfix 5787	. . . . . . . 8 ⊢ (a ∈ Fix ( S ∘ ImageR) ↔ a( S ∘ ImageR)a)
10		brco 4883	. . . . . . . . 9 ⊢ (a( S ∘ ImageR)a ↔ ∃b(aImageRb ∧ b S a))
11		vex 2862	. . . . . . . . . . . . 13 ⊢ b ∈ 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 ⊢ (∃b(aImageRb ∧ b S a) ↔ ∃b(b = (R “ a) ∧ b S a))
15		clos1ex.2	. . . . . . . . . . . 12 ⊢ R ∈ V
16	15, 6	imaex 4747	. . . . . . . . . . 11 ⊢ (R “ a) ∈ 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) ⊆ a)
19	17, 18	syl6bb 252	. . . . . . . . . . 11 ⊢ (b = (R “ a) → (b S a ↔ (R “ a) ⊆ a))
20	16, 19	ceqsexv 2894	. . . . . . . . . 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 2464	. . . 4 ⊢ (( S “ {S}) ∩ Fix ( S ∘ ImageR)) = {a ∣ (S ⊆ a ∧ (R “ a) ⊆ a)}
27		ssetex 4744	. . . . . 6 ⊢ S ∈ V
28		snex 4111	. . . . . 6 ⊢ {S} ∈ V
29	27, 28	imaex 4747	. . . . 5 ⊢ ( S “ {S}) ∈ V
30	15	imageex 5801	. . . . . . 7 ⊢ ImageR ∈ V
31	27, 30	coex 4750	. . . . . 6 ⊢ ( S ∘ ImageR) ∈ V
32	31	fixex 5789	. . . . 5 ⊢ Fix ( S ∘ ImageR) ∈ V
33	29, 32	inex 4105	. . . 4 ⊢ (( S “ {S}) ∩ Fix ( S ∘ ImageR)) ∈ V
34	26, 33	eqeltrri 2424	. . 3 ⊢ {a ∣ (S ⊆ a ∧ (R “ a) ⊆ a)} ∈ V
35	34	intex 4320	. 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 2859   ∩ cin 3208   ⊆ wss 3257  {csn 3737  ∩cint 3926  ⟨cop 4561   class class class wbr 4639   S csset 4719   ∘ ccom 4721   “ cima 4722   Fix cfix 5739  Imagecimage 5753   Clos1 cclos1 5872

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 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