MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r1elssi Unicode version

Theorem r1elssi 7493
Description: The range of the  R1 function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 7494 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 4142 . . . 4  |-  ( A. x  e.  On  Tr  ( R1 `  x )  ->  Tr  U_ x  e.  On  ( R1 `  x ) )
2 r1tr 7464 . . . . 5  |-  Tr  ( R1 `  x )
32a1i 10 . . . 4  |-  ( x  e.  On  ->  Tr  ( R1 `  x ) )
41, 3mprg 2625 . . 3  |-  Tr  U_ x  e.  On  ( R1 `  x )
5 r1funlim 7454 . . . . . 6  |-  ( Fun 
R1  /\  Lim  dom  R1 )
65simpli 444 . . . . 5  |-  Fun  R1
7 funiunfv 5790 . . . . 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 4135 . . . 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 4138 . 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 1632    e. wcel 1696    C_ wss 3165   U.cuni 3843   U_ciun 3921   Tr wtr 4129   Oncon0 4408   Lim wlim 4409   dom cdm 4705   "cima 4708   Fun wfun 5265   ` cfv 5271   R1cr1 7450
This theorem is referenced by:  r1elss  7494  pwwf  7495  rankelb  7512  rankval3b  7514  r1pw  7533  rankuni2b  7541  tcwf  7569  tcrank  7570  hsmexlem4  8071  rankcf  8415  wfgru  8454  grur1  8458
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439  df-r1 7452
  Copyright terms: Public domain W3C validator