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Theorem hsmexlem7 8755
Description: Lemma for hsmex 8764. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.)
Hypothesis
Ref Expression
hsmexlem7.h  |-  H  =  ( rec ( ( z  e.  _V  |->  (har
`  ~P ( X  X.  z ) ) ) ,  (har `  ~P X ) )  |`  om )
Assertion
Ref Expression
hsmexlem7  |-  ( H `
 (/) )  =  (har
`  ~P X )
Distinct variable group:    z, X
Allowed substitution hint:    H( z)

Proof of Theorem hsmexlem7
StepHypRef Expression
1 hsmexlem7.h . . 3  |-  H  =  ( rec ( ( z  e.  _V  |->  (har
`  ~P ( X  X.  z ) ) ) ,  (har `  ~P X ) )  |`  om )
21fveq1i 5806 . 2  |-  ( H `
 (/) )  =  ( ( rec ( ( z  e.  _V  |->  (har
`  ~P ( X  X.  z ) ) ) ,  (har `  ~P X ) )  |`  om ) `  (/) )
3 fvex 5815 . . 3  |-  (har `  ~P X )  e.  _V
4 fr0g 7058 . . 3  |-  ( (har
`  ~P X )  e.  _V  ->  (
( rec ( ( z  e.  _V  |->  (har
`  ~P ( X  X.  z ) ) ) ,  (har `  ~P X ) )  |`  om ) `  (/) )  =  (har `  ~P X ) )
53, 4ax-mp 5 . 2  |-  ( ( rec ( ( z  e.  _V  |->  (har `  ~P ( X  X.  z
) ) ) ,  (har `  ~P X ) )  |`  om ) `  (/) )  =  (har
`  ~P X )
62, 5eqtri 2431 1  |-  ( H `
 (/) )  =  (har
`  ~P X )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405    e. wcel 1842   _Vcvv 3058   (/)c0 3737   ~Pcpw 3954    |-> cmpt 4452    X. cxp 4940    |` cres 4944   ` cfv 5525   omcom 6638   reccrdg 7032  harchar 7936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-om 6639  df-recs 6999  df-rdg 7033
This theorem is referenced by:  hsmexlem9  8757  hsmexlem4  8761
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