MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tz9.12lem2 Unicode version

Theorem tz9.12lem2 7649
Description: Lemma for tz9.12 7651. (Contributed by NM, 22-Sep-2003.)
Hypotheses
Ref Expression
tz9.12lem.1  |-  A  e. 
_V
tz9.12lem.2  |-  F  =  ( z  e.  _V  |->  |^|
{ v  e.  On  |  z  e.  ( R1 `  v ) } )
Assertion
Ref Expression
tz9.12lem2  |-  suc  U. ( F " A )  e.  On
Distinct variable group:    z, v, A
Allowed substitution hints:    F( z, v)

Proof of Theorem tz9.12lem2
StepHypRef Expression
1 tz9.12lem.1 . . . 4  |-  A  e. 
_V
2 tz9.12lem.2 . . . 4  |-  F  =  ( z  e.  _V  |->  |^|
{ v  e.  On  |  z  e.  ( R1 `  v ) } )
31, 2tz9.12lem1 7648 . . 3  |-  ( F
" A )  C_  On
42funmpt2 5432 . . . . 5  |-  Fun  F
51funimaex 5473 . . . . 5  |-  ( Fun 
F  ->  ( F " A )  e.  _V )
64, 5ax-mp 8 . . . 4  |-  ( F
" A )  e. 
_V
76ssonunii 4710 . . 3  |-  ( ( F " A ) 
C_  On  ->  U. ( F " A )  e.  On )
83, 7ax-mp 8 . 2  |-  U. ( F " A )  e.  On
98onsuci 4760 1  |-  suc  U. ( F " A )  e.  On
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717   {crab 2655   _Vcvv 2901    C_ wss 3265   U.cuni 3959   |^|cint 3994    e. cmpt 4209   Oncon0 4524   suc csuc 4526   "cima 4823   Fun wfun 5390   ` cfv 5396   R1cr1 7623
This theorem is referenced by:  tz9.12lem3  7650  tz9.12  7651
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-suc 4530  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-fun 5398
  Copyright terms: Public domain W3C validator