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Theorem jech9.3 7454
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 7407 . . 3  |-  R1  Fn  On
2 fniunfv 5707 . . 3  |-  ( R1  Fn  On  ->  U_ x  e.  On  ( R1 `  x )  =  U. ran  R1 )
31, 2ax-mp 10 . 2  |-  U_ x  e.  On  ( R1 `  x )  =  U. ran  R1
4 fndm 5281 . . . . . 6  |-  ( R1  Fn  On  ->  dom  R1  =  On )
51, 4ax-mp 10 . . . . 5  |-  dom  R1  =  On
65imaeq2i 4998 . . . 4  |-  ( R1
" dom  R1 )  =  ( R1 " On )
7 imadmrn 5012 . . . 4  |-  ( R1
" dom  R1 )  =  ran  R1
86, 7eqtr3i 2280 . . 3  |-  ( R1
" On )  =  ran  R1
98unieqi 3811 . 2  |-  U. ( R1 " On )  = 
U. ran  R1
10 unir1 7453 . 2  |-  U. ( R1 " On )  =  _V
113, 9, 103eqtr2i 2284 1  |-  U_ x  e.  On  ( R1 `  x )  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1619   _Vcvv 2763   U.cuni 3801   U_ciun 3879   Oncon0 4364   dom cdm 4661   ran crn 4662   "cima 4664    Fn wfn 4668   ` cfv 4673   R1cr1 7402
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-reg 7274  ax-inf2 7310
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-recs 6356  df-rdg 6391  df-r1 7404
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