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

Theorem tz9.12lem2 7476
Description: Lemma for tz9.12 7478. (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 7475 . . 3  |-  ( F
" A )  C_  On
42funmpt2 5307 . . . . 5  |-  Fun  F
51funimaex 5346 . . . . 5  |-  ( Fun 
F  ->  ( F " A )  e.  _V )
64, 5ax-mp 8 . . . 4  |-  ( F
" A )  e. 
_V
76ssonunii 4595 . . 3  |-  ( ( F " A ) 
C_  On  ->  U. ( F " A )  e.  On )
83, 7ax-mp 8 . 2  |-  U. ( F " A )  e.  On
98onsuci 4645 1  |-  suc  U. ( F " A )  e.  On
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    C_ wss 3165   U.cuni 3843   |^|cint 3878    e. cmpt 4093   Oncon0 4408   suc csuc 4410   "cima 4708   Fun wfun 5265   ` cfv 5271   R1cr1 7450
This theorem is referenced by:  tz9.12lem3  7477  tz9.12  7478
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273
  Copyright terms: Public domain W3C validator