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Theorem r1fnon 7455
Description: The cumulative hierarchy of sets function is a function on the class of ordinal numbers. (Contributed by NM, 5-Oct-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)
Assertion
Ref Expression
r1fnon  |-  R1  Fn  On

Proof of Theorem r1fnon
StepHypRef Expression
1 rdgfnon 6447 . 2  |-  rec (
( x  e.  _V  |->  ~P x ) ,  (/) )  Fn  On
2 df-r1 7452 . . 3  |-  R1  =  rec ( ( x  e. 
_V  |->  ~P x ) ,  (/) )
32fneq1i 5354 . 2  |-  ( R1  Fn  On  <->  rec (
( x  e.  _V  |->  ~P x ) ,  (/) )  Fn  On )
41, 3mpbir 200 1  |-  R1  Fn  On
Colors of variables: wff set class
Syntax hints:   _Vcvv 2801   (/)c0 3468   ~Pcpw 3638    e. cmpt 4093   Oncon0 4408    Fn wfn 5266   reccrdg 6438   R1cr1 7450
This theorem is referenced by:  r1suc  7458  r1lim  7460  r111  7463  r1ord  7468  r1ord3  7470  r1elss  7494  jech9.3  7502  onwf  7518  ssrankr1  7523  r1val3  7526  r1pw  7533  rankuni  7551  rankr1b  7552  r1om  7886  hsmexlem6  8073  smobeth  8224  wunr1om  8357  r1limwun  8374  r1wunlim  8375  tskr1om  8405  tskr1om2  8406  inar1  8413  rankcf  8415  inatsk  8416  r1tskina  8420  grur1  8458  grothomex  8467  aomclem4  27257
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-pow 4204  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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-iun 3923  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-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439  df-r1 7452
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