MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hsmexlem7 Structured version   Unicode version

Theorem hsmexlem7 8696
Description: Lemma for hsmex 8705. 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 5793 . 2  |-  ( H `
 (/) )  =  ( ( rec ( ( z  e.  _V  |->  (har
`  ~P ( X  X.  z ) ) ) ,  (har `  ~P X ) )  |`  om ) `  (/) )
3 fvex 5802 . . 3  |-  (har `  ~P X )  e.  _V
4 fr0g 6994 . . 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 2480 1  |-  ( H `
 (/) )  =  (har
`  ~P X )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758   _Vcvv 3071   (/)c0 3738   ~Pcpw 3961    |-> cmpt 4451    X. cxp 4939    |` cres 4943   ` cfv 5519   omcom 6579   reccrdg 6968  harchar 7875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-om 6580  df-recs 6935  df-rdg 6969
This theorem is referenced by:  hsmexlem9  8698  hsmexlem4  8702
  Copyright terms: Public domain W3C validator