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Theorem jech9.3 7488
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 7441 . . 3  |-  R1  Fn  On
2 fniunfv 5775 . . 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 5345 . . . . . 6  |-  ( R1  Fn  On  ->  dom  R1  =  On )
51, 4ax-mp 8 . . . . 5  |-  dom  R1  =  On
65imaeq2i 5012 . . . 4  |-  ( R1
" dom  R1 )  =  ( R1 " On )
7 imadmrn 5026 . . . 4  |-  ( R1
" dom  R1 )  =  ran  R1
86, 7eqtr3i 2307 . . 3  |-  ( R1
" On )  =  ran  R1
98unieqi 3839 . 2  |-  U. ( R1 " On )  = 
U. ran  R1
10 unir1 7487 . 2  |-  U. ( R1 " On )  =  _V
113, 9, 103eqtr2i 2311 1  |-  U_ x  e.  On  ( R1 `  x )  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1625   _Vcvv 2790   U.cuni 3829   U_ciun 3907   Oncon0 4394   dom cdm 4691   ran crn 4692   "cima 4694    Fn wfn 5252   ` cfv 5257   R1cr1 7436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-reg 7308  ax-inf2 7344
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-recs 6390  df-rdg 6425  df-r1 7438
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