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Theorem jech9.3 8017
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 7970 . . 3  |-  R1  Fn  On
2 fniunfv 5961 . . 3  |-  ( R1  Fn  On  ->  U_ x  e.  On  ( R1 `  x )  =  U. ran  R1 )
31, 2ax-mp 5 . 2  |-  U_ x  e.  On  ( R1 `  x )  =  U. ran  R1
4 fndm 5507 . . . . . 6  |-  ( R1  Fn  On  ->  dom  R1  =  On )
51, 4ax-mp 5 . . . . 5  |-  dom  R1  =  On
65imaeq2i 5164 . . . 4  |-  ( R1
" dom  R1 )  =  ( R1 " On )
7 imadmrn 5176 . . . 4  |-  ( R1
" dom  R1 )  =  ran  R1
86, 7eqtr3i 2463 . . 3  |-  ( R1
" On )  =  ran  R1
98unieqi 4097 . 2  |-  U. ( R1 " On )  = 
U. ran  R1
10 unir1 8016 . 2  |-  U. ( R1 " On )  =  _V
113, 9, 103eqtr2i 2467 1  |-  U_ x  e.  On  ( R1 `  x )  =  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1364   _Vcvv 2970   U.cuni 4088   U_ciun 4168   Oncon0 4715   dom cdm 4836   ran crn 4837   "cima 4839    Fn wfn 5410   ` cfv 5415   R1cr1 7965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-reg 7803  ax-inf2 7843
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-om 6476  df-recs 6828  df-rdg 6862  df-r1 7967
This theorem is referenced by: (None)
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