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Theorem r1elssi 7477
Description: The range of the  R1 function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 7478 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 4126 . . . 4  |-  ( A. x  e.  On  Tr  ( R1 `  x )  ->  Tr  U_ x  e.  On  ( R1 `  x ) )
2 r1tr 7448 . . . . 5  |-  Tr  ( R1 `  x )
32a1i 10 . . . 4  |-  ( x  e.  On  ->  Tr  ( R1 `  x ) )
41, 3mprg 2612 . . 3  |-  Tr  U_ x  e.  On  ( R1 `  x )
5 r1funlim 7438 . . . . . 6  |-  ( Fun 
R1  /\  Lim  dom  R1 )
65simpli 444 . . . . 5  |-  Fun  R1
7 funiunfv 5774 . . . . 5  |-  ( Fun 
R1  ->  U_ x  e.  On  ( R1 `  x )  =  U. ( R1
" On ) )
86, 7ax-mp 8 . . . 4  |-  U_ x  e.  On  ( R1 `  x )  =  U. ( R1 " On )
9 treq 4119 . . . 4  |-  ( U_ x  e.  On  ( R1 `  x )  = 
U. ( R1 " On )  ->  ( Tr 
U_ x  e.  On  ( R1 `  x )  <->  Tr  U. ( R1 " On ) ) )
108, 9ax-mp 8 . . 3  |-  ( Tr 
U_ x  e.  On  ( R1 `  x )  <->  Tr  U. ( R1 " On ) )
114, 10mpbi 199 . 2  |-  Tr  U. ( R1 " On )
12 trss 4122 . 2  |-  ( Tr 
U. ( R1 " On )  ->  ( A  e.  U. ( R1
" On )  ->  A  C_  U. ( R1
" On ) ) )
1311, 12ax-mp 8 1  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  U. ( R1
" On ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684    C_ wss 3152   U.cuni 3827   U_ciun 3905   Tr wtr 4113   Oncon0 4392   Lim wlim 4393   dom cdm 4689   "cima 4692   Fun wfun 5249   ` cfv 5255   R1cr1 7434
This theorem is referenced by:  r1elss  7478  pwwf  7479  rankelb  7496  rankval3b  7498  r1pw  7517  rankuni2b  7525  tcwf  7553  tcrank  7554  hsmexlem4  8055  rankcf  8399  wfgru  8438  grur1  8442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423  df-r1 7436
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