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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.
StepHypRef Expression
1 df-clos1 5873 . 2 Clos1 (S, R) = {a (S a (Ra) 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 aS, a S )
5 clos1ex.1 . . . . . . . . 9 S V
6 vex 2862 . . . . . . . . 9 a V
75, 6brsset 4758 . . . . . . . 8 (S S aS a)
83, 4, 73bitr2i 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)ab(aImageRb b S a))
11 vex 2862 . . . . . . . . . . . . 13 b V
126, 11brimage 5793 . . . . . . . . . . . 12 (aImageRbb = (Ra))
1312anbi1i 676 . . . . . . . . . . 11 ((aImageRb b S a) ↔ (b = (Ra) b S a))
1413exbii 1582 . . . . . . . . . 10 (b(aImageRb b S a) ↔ b(b = (Ra) b S a))
15 clos1ex.2 . . . . . . . . . . . 12 R V
1615, 6imaex 4747 . . . . . . . . . . 11 (Ra) V
17 breq1 4642 . . . . . . . . . . . 12 (b = (Ra) → (b S a ↔ (Ra) S a))
1816, 6brsset 4758 . . . . . . . . . . . 12 ((Ra) S a ↔ (Ra) a)
1917, 18syl6bb 252 . . . . . . . . . . 11 (b = (Ra) → (b S a ↔ (Ra) a))
2016, 19ceqsexv 2894 . . . . . . . . . 10 (b(b = (Ra) b S a) ↔ (Ra) a)
2114, 20bitri 240 . . . . . . . . 9 (b(aImageRb b S a) ↔ (Ra) a)
2210, 21bitri 240 . . . . . . . 8 (a( S ImageR)a ↔ (Ra) a)
239, 22bitri 240 . . . . . . 7 (a Fix ( S ImageR) ↔ (Ra) a)
248, 23anbi12i 678 . . . . . 6 ((a ( S “ {S}) a Fix ( S ImageR)) ↔ (S a (Ra) a))
252, 24bitri 240 . . . . 5 (a (( S “ {S}) ∩ Fix ( S ImageR)) ↔ (S a (Ra) a))
2625abbi2i 2464 . . . 4 (( S “ {S}) ∩ Fix ( S ImageR)) = {a (S a (Ra) a)}
27 ssetex 4744 . . . . . 6 S V
28 snex 4111 . . . . . 6 {S} V
2927, 28imaex 4747 . . . . 5 ( S “ {S}) V
3015imageex 5801 . . . . . . 7 ImageR V
3127, 30coex 4750 . . . . . 6 ( S ImageR) V
3231fixex 5789 . . . . 5 Fix ( S ImageR) V
3329, 32inex 4105 . . . 4 (( S “ {S}) ∩ Fix ( S ImageR)) V
3426, 33eqeltrri 2424 . . 3 {a (S a (Ra) a)} V
3534intex 4320 . 2 {a (S a (Ra) a)} V
361, 35eqeltri 2423 1 Clos1 (S, R) V
 Colors of variables: wff setvar class 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-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
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