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Theorem r1fnon 8078
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 6977 . 2  |-  rec (
( x  e.  _V  |->  ~P x ) ,  (/) )  Fn  On
2 df-r1 8075 . . 3  |-  R1  =  rec ( ( x  e. 
_V  |->  ~P x ) ,  (/) )
32fneq1i 5606 . 2  |-  ( R1  Fn  On  <->  rec (
( x  e.  _V  |->  ~P x ) ,  (/) )  Fn  On )
41, 3mpbir 209 1  |-  R1  Fn  On
Colors of variables: wff setvar class
Syntax hints:   _Vcvv 3071   (/)c0 3738   ~Pcpw 3961    |-> cmpt 4451   Oncon0 4820    Fn wfn 5514   reccrdg 6968   R1cr1 8073
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-rep 4504  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-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-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-recs 6935  df-rdg 6969  df-r1 8075
This theorem is referenced by:  r1suc  8081  r1lim  8083  r111  8086  r1ord  8091  r1ord3  8093  r1elss  8117  jech9.3  8125  onwf  8141  ssrankr1  8146  r1val3  8149  r1pw  8156  rankuni  8174  rankr1b  8175  r1om  8517  hsmexlem6  8704  smobeth  8854  wunr1om  8990  r1limwun  9007  r1wunlim  9008  tskr1om  9038  tskr1om2  9039  inar1  9046  rankcf  9048  inatsk  9049  r1tskina  9053  grur1  9091  grothomex  9100  aomclem4  29551
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