MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r1fnon Structured version   Unicode version

Theorem r1fnon 7685
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 6668 . 2  |-  rec (
( x  e.  _V  |->  ~P x ) ,  (/) )  Fn  On
2 df-r1 7682 . . 3  |-  R1  =  rec ( ( x  e. 
_V  |->  ~P x ) ,  (/) )
32fneq1i 5531 . 2  |-  ( R1  Fn  On  <->  rec (
( x  e.  _V  |->  ~P x ) ,  (/) )  Fn  On )
41, 3mpbir 201 1  |-  R1  Fn  On
Colors of variables: wff set class
Syntax hints:   _Vcvv 2948   (/)c0 3620   ~Pcpw 3791    e. cmpt 4258   Oncon0 4573    Fn wfn 5441   reccrdg 6659   R1cr1 7680
This theorem is referenced by:  r1suc  7688  r1lim  7690  r111  7693  r1ord  7698  r1ord3  7700  r1elss  7724  jech9.3  7732  onwf  7748  ssrankr1  7753  r1val3  7756  r1pw  7763  rankuni  7781  rankr1b  7782  r1om  8116  hsmexlem6  8303  smobeth  8453  wunr1om  8586  r1limwun  8603  r1wunlim  8604  tskr1om  8634  tskr1om2  8635  inar1  8642  rankcf  8644  inatsk  8645  r1tskina  8649  grur1  8687  grothomex  8696  aomclem4  27123
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-pow 4369  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-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  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-iun 4087  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-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-recs 6625  df-rdg 6660  df-r1 7682
  Copyright terms: Public domain W3C validator