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Theorem r1elssi 8017
Description: The range of the  R1 function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 8018 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 4403 . . . 4  |-  ( A. x  e.  On  Tr  ( R1 `  x )  ->  Tr  U_ x  e.  On  ( R1 `  x ) )
2 r1tr 7988 . . . . 5  |-  Tr  ( R1 `  x )
32a1i 11 . . . 4  |-  ( x  e.  On  ->  Tr  ( R1 `  x ) )
41, 3mprg 2790 . . 3  |-  Tr  U_ x  e.  On  ( R1 `  x )
5 r1funlim 7978 . . . . . 6  |-  ( Fun 
R1  /\  Lim  dom  R1 )
65simpli 458 . . . . 5  |-  Fun  R1
7 funiunfv 5970 . . . . 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 4396 . . . 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 4399 . 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 1369    e. wcel 1756    C_ wss 3333   U.cuni 4096   U_ciun 4176   Tr wtr 4390   Oncon0 4724   Lim wlim 4725   dom cdm 4845   "cima 4848   Fun wfun 5417   ` cfv 5423   R1cr1 7974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-om 6482  df-recs 6837  df-rdg 6871  df-r1 7976
This theorem is referenced by:  r1elss  8018  pwwf  8019  rankelb  8036  rankval3b  8038  r1pw  8057  rankuni2b  8065  tcwf  8095  tcrank  8096  hsmexlem4  8603  rankcf  8949  wfgru  8988  grur1  8992
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