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Theorem clos1induct 5880
 Description: Inductive law for closure. If the base set is a subset of X, and X is closed under R, then the closure is a subset of X. Theorem IX.5.15 of [Rosser] p. 247. (Contributed by SF, 11-Feb-2015.)
Hypotheses
Ref Expression
clos1induct.1 S V
clos1induct.2 R V
clos1induct.3 C = Clos1 (S, R)
Assertion
Ref Expression
clos1induct ((X V S X x C z((x X xRz) → z X)) → C X)
Distinct variable groups:   x,C,z   x,R,z   x,X,z
Allowed substitution hints:   S(x,z)   V(x,z)

Proof of Theorem clos1induct
Dummy variable a is distinct from all other variables.
StepHypRef Expression
1 clos1induct.3 . . . 4 C = Clos1 (S, R)
2 clos1induct.1 . . . . 5 S V
3 clos1induct.2 . . . . 5 R V
42, 3clos1ex 5876 . . . 4 Clos1 (S, R) V
51, 4eqeltri 2423 . . 3 C V
6 inexg 4100 . . 3 ((X V C V) → (XC) V)
75, 6mpan2 652 . 2 (X V → (XC) V)
81clos1base 5878 . . 3 S C
9 ssin 3477 . . . 4 ((S X S C) ↔ S (XC))
109biimpi 186 . . 3 ((S X S C) → S (XC))
118, 10mpan2 652 . 2 (S XS (XC))
12 elima2 4755 . . . . . . 7 (z (R “ (XC)) ↔ x(x (XC) xRz))
13 elin 3219 . . . . . . 7 (z (XC) ↔ (z X z C))
1412, 13imbi12i 316 . . . . . 6 ((z (R “ (XC)) → z (XC)) ↔ (x(x (XC) xRz) → (z X z C)))
15 df-ral 2619 . . . . . . . 8 (x C ((x X xRz) → z X) ↔ x(x C → ((x X xRz) → z X)))
16 impexp 433 . . . . . . . . . 10 (((x C (x X xRz)) → z X) ↔ (x C → ((x X xRz) → z X)))
171clos1conn 5879 . . . . . . . . . . . . 13 ((x C xRz) → z C)
1817biantrud 493 . . . . . . . . . . . 12 ((x C xRz) → (z X ↔ (z X z C)))
1918adantrl 696 . . . . . . . . . . 11 ((x C (x X xRz)) → (z X ↔ (z X z C)))
2019pm5.74i 236 . . . . . . . . . 10 (((x C (x X xRz)) → z X) ↔ ((x C (x X xRz)) → (z X z C)))
2116, 20bitr3i 242 . . . . . . . . 9 ((x C → ((x X xRz) → z X)) ↔ ((x C (x X xRz)) → (z X z C)))
2221albii 1566 . . . . . . . 8 (x(x C → ((x X xRz) → z X)) ↔ x((x C (x X xRz)) → (z X z C)))
2315, 22bitri 240 . . . . . . 7 (x C ((x X xRz) → z X) ↔ x((x C (x X xRz)) → (z X z C)))
24 elin 3219 . . . . . . . . . . . 12 (x (XC) ↔ (x X x C))
25 ancom 437 . . . . . . . . . . . 12 ((x X x C) ↔ (x C x X))
2624, 25bitri 240 . . . . . . . . . . 11 (x (XC) ↔ (x C x X))
2726anbi1i 676 . . . . . . . . . 10 ((x (XC) xRz) ↔ ((x C x X) xRz))
28 anass 630 . . . . . . . . . 10 (((x C x X) xRz) ↔ (x C (x X xRz)))
2927, 28bitri 240 . . . . . . . . 9 ((x (XC) xRz) ↔ (x C (x X xRz)))
3029imbi1i 315 . . . . . . . 8 (((x (XC) xRz) → (z X z C)) ↔ ((x C (x X xRz)) → (z X z C)))
3130albii 1566 . . . . . . 7 (x((x (XC) xRz) → (z X z C)) ↔ x((x C (x X xRz)) → (z X z C)))
32 19.23v 1891 . . . . . . 7 (x((x (XC) xRz) → (z X z C)) ↔ (x(x (XC) xRz) → (z X z C)))
3323, 31, 323bitr2i 264 . . . . . 6 (x C ((x X xRz) → z X) ↔ (x(x (XC) xRz) → (z X z C)))
3414, 33bitr4i 243 . . . . 5 ((z (R “ (XC)) → z (XC)) ↔ x C ((x X xRz) → z X))
3534albii 1566 . . . 4 (z(z (R “ (XC)) → z (XC)) ↔ zx C ((x X xRz) → z X))
36 dfss2 3262 . . . 4 ((R “ (XC)) (XC) ↔ z(z (R “ (XC)) → z (XC)))
37 ralcom4 2877 . . . 4 (x C z((x X xRz) → z X) ↔ zx C ((x X xRz) → z X))
3835, 36, 373bitr4i 268 . . 3 ((R “ (XC)) (XC) ↔ x C z((x X xRz) → z X))
3938biimpri 197 . 2 (x C z((x X xRz) → z X) → (R “ (XC)) (XC))
40 df-clos1 5873 . . . . 5 Clos1 (S, R) = {a (S a (Ra) a)}
411, 40eqtri 2373 . . . 4 C = {a (S a (Ra) a)}
42 sseq2 3293 . . . . . . . . 9 (a = (XC) → (S aS (XC)))
43 imaeq2 4938 . . . . . . . . . 10 (a = (XC) → (Ra) = (R “ (XC)))
44 id 19 . . . . . . . . . 10 (a = (XC) → a = (XC))
4543, 44sseq12d 3300 . . . . . . . . 9 (a = (XC) → ((Ra) a ↔ (R “ (XC)) (XC)))
4642, 45anbi12d 691 . . . . . . . 8 (a = (XC) → ((S a (Ra) a) ↔ (S (XC) (R “ (XC)) (XC))))
4746elabg 2986 . . . . . . 7 ((XC) V → ((XC) {a (S a (Ra) a)} ↔ (S (XC) (R “ (XC)) (XC))))
4847biimprd 214 . . . . . 6 ((XC) V → ((S (XC) (R “ (XC)) (XC)) → (XC) {a (S a (Ra) a)}))
49483impib 1149 . . . . 5 (((XC) V S (XC) (R “ (XC)) (XC)) → (XC) {a (S a (Ra) a)})
50 intss1 3941 . . . . 5 ((XC) {a (S a (Ra) a)} → {a (S a (Ra) a)} (XC))
5149, 50syl 15 . . . 4 (((XC) V S (XC) (R “ (XC)) (XC)) → {a (S a (Ra) a)} (XC))
5241, 51syl5eqss 3315 . . 3 (((XC) V S (XC) (R “ (XC)) (XC)) → C (XC))
53 inss1 3475 . . 3 (XC) X
5452, 53syl6ss 3284 . 2 (((XC) V S (XC) (R “ (XC)) (XC)) → C X)
557, 11, 39, 54syl3an 1224 1 ((X V S X x C z((x X xRz) → z X)) → C X)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614  Vcvv 2859   ∩ cin 3208   ⊆ wss 3257  ∩cint 3926   class class class wbr 4639   “ cima 4722   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:  clos1is  5881  clos1nrel  5886  clos10  5887  spacind  6287  frecxp  6314
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