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Theorem jech9.3 8223
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 8176 . . 3  |-  R1  Fn  On
2 fniunfv 6140 . . 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 5673 . . . . . 6  |-  ( R1  Fn  On  ->  dom  R1  =  On )
51, 4ax-mp 5 . . . . 5  |-  dom  R1  =  On
65imaeq2i 5328 . . . 4  |-  ( R1
" dom  R1 )  =  ( R1 " On )
7 imadmrn 5340 . . . 4  |-  ( R1
" dom  R1 )  =  ran  R1
86, 7eqtr3i 2493 . . 3  |-  ( R1
" On )  =  ran  R1
98unieqi 4249 . 2  |-  U. ( R1 " On )  = 
U. ran  R1
10 unir1 8222 . 2  |-  U. ( R1 " On )  =  _V
113, 9, 103eqtr2i 2497 1  |-  U_ x  e.  On  ( R1 `  x )  =  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374   _Vcvv 3108   U.cuni 4240   U_ciun 4320   Oncon0 4873   dom cdm 4994   ran crn 4995   "cima 4997    Fn wfn 5576   ` cfv 5581   R1cr1 8171
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-reg 8009  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-om 6674  df-recs 7034  df-rdg 7068  df-r1 8173
This theorem is referenced by: (None)
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