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Theorem r1elssi 8235
Description: The range of the  R1 function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 8236 that doesn't need  A to be a set. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
r1elssi  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  U. ( R1
" On ) )

Proof of Theorem r1elssi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 triun 4559 . . . 4  |-  ( A. x  e.  On  Tr  ( R1 `  x )  ->  Tr  U_ x  e.  On  ( R1 `  x ) )
2 r1tr 8206 . . . . 5  |-  Tr  ( R1 `  x )
32a1i 11 . . . 4  |-  ( x  e.  On  ->  Tr  ( R1 `  x ) )
41, 3mprg 2830 . . 3  |-  Tr  U_ x  e.  On  ( R1 `  x )
5 r1funlim 8196 . . . . . 6  |-  ( Fun 
R1  /\  Lim  dom  R1 )
65simpli 458 . . . . 5  |-  Fun  R1
7 funiunfv 6159 . . . . 5  |-  ( Fun 
R1  ->  U_ x  e.  On  ( R1 `  x )  =  U. ( R1
" On ) )
86, 7ax-mp 5 . . . 4  |-  U_ x  e.  On  ( R1 `  x )  =  U. ( R1 " On )
9 treq 4552 . . . 4  |-  ( U_ x  e.  On  ( R1 `  x )  = 
U. ( R1 " On )  ->  ( Tr 
U_ x  e.  On  ( R1 `  x )  <->  Tr  U. ( R1 " On ) ) )
108, 9ax-mp 5 . . 3  |-  ( Tr 
U_ x  e.  On  ( R1 `  x )  <->  Tr  U. ( R1 " On ) )
114, 10mpbi 208 . 2  |-  Tr  U. ( R1 " On )
12 trss 4555 . 2  |-  ( Tr 
U. ( R1 " On )  ->  ( A  e.  U. ( R1
" On )  ->  A  C_  U. ( R1
" On ) ) )
1311, 12ax-mp 5 1  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  U. ( R1
" On ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767    C_ wss 3481   U.cuni 4251   U_ciun 4331   Tr wtr 4546   Oncon0 4884   Lim wlim 4885   dom cdm 5005   "cima 5008   Fun wfun 5588   ` cfv 5594   R1cr1 8192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6696  df-recs 7054  df-rdg 7088  df-r1 8194
This theorem is referenced by:  r1elss  8236  pwwf  8237  rankelb  8254  rankval3b  8256  r1pw  8275  rankuni2b  8283  tcwf  8313  tcrank  8314  hsmexlem4  8821  rankcf  9167  wfgru  9206  grur1  9210
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