MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  jech9.3 Unicode version

Theorem jech9.3 7486
Description: Every set belongs to some value of the cumulative hierarchy of sets function  R1, i.e. the indexed union of all values of 
R1 is the universe. Lemma 9.3 of [Jech] p. 71. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 8-Jun-2013.)
Assertion
Ref Expression
jech9.3  |-  U_ x  e.  On  ( R1 `  x )  =  _V

Proof of Theorem jech9.3
StepHypRef Expression
1 r1fnon 7439 . . 3  |-  R1  Fn  On
2 fniunfv 5773 . . 3  |-  ( R1  Fn  On  ->  U_ x  e.  On  ( R1 `  x )  =  U. ran  R1 )
31, 2ax-mp 8 . 2  |-  U_ x  e.  On  ( R1 `  x )  =  U. ran  R1
4 fndm 5343 . . . . . 6  |-  ( R1  Fn  On  ->  dom  R1  =  On )
51, 4ax-mp 8 . . . . 5  |-  dom  R1  =  On
65imaeq2i 5010 . . . 4  |-  ( R1
" dom  R1 )  =  ( R1 " On )
7 imadmrn 5024 . . . 4  |-  ( R1
" dom  R1 )  =  ran  R1
86, 7eqtr3i 2305 . . 3  |-  ( R1
" On )  =  ran  R1
98unieqi 3837 . 2  |-  U. ( R1 " On )  = 
U. ran  R1
10 unir1 7485 . 2  |-  U. ( R1 " On )  =  _V
113, 9, 103eqtr2i 2309 1  |-  U_ x  e.  On  ( R1 `  x )  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1623   _Vcvv 2788   U.cuni 3827   U_ciun 3905   Oncon0 4392   dom cdm 4689   ran crn 4690   "cima 4692    Fn wfn 5250   ` cfv 5255   R1cr1 7434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7306  ax-inf2 7342
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423  df-r1 7436
  Copyright terms: Public domain W3C validator