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Theorem tz9.12lem2 7706
Description: Lemma for tz9.12 7708. (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 7705 . . 3  |-  ( F
" A )  C_  On
42funmpt2 5482 . . . . 5  |-  Fun  F
51funimaex 5523 . . . . 5  |-  ( Fun 
F  ->  ( F " A )  e.  _V )
64, 5ax-mp 8 . . . 4  |-  ( F
" A )  e. 
_V
76ssonunii 4760 . . 3  |-  ( ( F " A ) 
C_  On  ->  U. ( F " A )  e.  On )
83, 7ax-mp 8 . 2  |-  U. ( F " A )  e.  On
98onsuci 4810 1  |-  suc  U. ( F " A )  e.  On
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   {crab 2701   _Vcvv 2948    C_ wss 3312   U.cuni 4007   |^|cint 4042    e. cmpt 4258   Oncon0 4573   suc csuc 4575   "cima 4873   Fun wfun 5440   ` cfv 5446   R1cr1 7680
This theorem is referenced by:  tz9.12lem3  7707  tz9.12  7708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-fun 5448
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