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Theorem r1elssi 7715
Description: The range of the  R1 function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 7716 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 4302 . . . 4  |-  ( A. x  e.  On  Tr  ( R1 `  x )  ->  Tr  U_ x  e.  On  ( R1 `  x ) )
2 r1tr 7686 . . . . 5  |-  Tr  ( R1 `  x )
32a1i 11 . . . 4  |-  ( x  e.  On  ->  Tr  ( R1 `  x ) )
41, 3mprg 2762 . . 3  |-  Tr  U_ x  e.  On  ( R1 `  x )
5 r1funlim 7676 . . . . . 6  |-  ( Fun 
R1  /\  Lim  dom  R1 )
65simpli 445 . . . . 5  |-  Fun  R1
7 funiunfv 5981 . . . . 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 4295 . . . 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 200 . 2  |-  Tr  U. ( R1 " On )
12 trss 4298 . 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 177    = wceq 1652    e. wcel 1725    C_ wss 3307   U.cuni 4002   U_ciun 4080   Tr wtr 4289   Oncon0 4568   Lim wlim 4569   dom cdm 4864   "cima 4867   Fun wfun 5434   ` cfv 5440   R1cr1 7672
This theorem is referenced by:  r1elss  7716  pwwf  7717  rankelb  7734  rankval3b  7736  r1pw  7755  rankuni2b  7763  tcwf  7791  tcrank  7792  hsmexlem4  8293  rankcf  8636  wfgru  8675  grur1  8679
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-reu 2699  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-recs 6619  df-rdg 6654  df-r1 7674
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