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Theorem r1elss 7688
Description: The range of the  R1 function is transitive. Lemma 2.10 of [Kunen] p. 97. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
Hypothesis
Ref Expression
r1elss.1  |-  A  e. 
_V
Assertion
Ref Expression
r1elss  |-  ( A  e.  U. ( R1
" On )  <->  A  C_  U. ( R1 " On ) )

Proof of Theorem r1elss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r1elssi 7687 . 2  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  U. ( R1
" On ) )
2 r1elss.1 . . . 4  |-  A  e. 
_V
32tz9.12 7672 . . 3  |-  ( A. y  e.  A  E. x  e.  On  y  e.  ( R1 `  x
)  ->  E. x  e.  On  A  e.  ( R1 `  x ) )
4 dfss3 3298 . . . 4  |-  ( A 
C_  U. ( R1 " On )  <->  A. y  e.  A  y  e.  U. ( R1 " On ) )
5 r1fnon 7649 . . . . . . . 8  |-  R1  Fn  On
6 fnfun 5501 . . . . . . . 8  |-  ( R1  Fn  On  ->  Fun  R1 )
7 funiunfv 5954 . . . . . . . 8  |-  ( Fun 
R1  ->  U_ x  e.  On  ( R1 `  x )  =  U. ( R1
" On ) )
85, 6, 7mp2b 10 . . . . . . 7  |-  U_ x  e.  On  ( R1 `  x )  =  U. ( R1 " On )
98eleq2i 2468 . . . . . 6  |-  ( y  e.  U_ x  e.  On  ( R1 `  x )  <->  y  e.  U. ( R1 " On ) )
10 eliun 4057 . . . . . 6  |-  ( y  e.  U_ x  e.  On  ( R1 `  x )  <->  E. x  e.  On  y  e.  ( R1 `  x ) )
119, 10bitr3i 243 . . . . 5  |-  ( y  e.  U. ( R1
" On )  <->  E. x  e.  On  y  e.  ( R1 `  x ) )
1211ralbii 2690 . . . 4  |-  ( A. y  e.  A  y  e.  U. ( R1 " On )  <->  A. y  e.  A  E. x  e.  On  y  e.  ( R1 `  x ) )
134, 12bitri 241 . . 3  |-  ( A 
C_  U. ( R1 " On )  <->  A. y  e.  A  E. x  e.  On  y  e.  ( R1 `  x ) )
148eleq2i 2468 . . . 4  |-  ( A  e.  U_ x  e.  On  ( R1 `  x )  <->  A  e.  U. ( R1 " On ) )
15 eliun 4057 . . . 4  |-  ( A  e.  U_ x  e.  On  ( R1 `  x )  <->  E. x  e.  On  A  e.  ( R1 `  x ) )
1614, 15bitr3i 243 . . 3  |-  ( A  e.  U. ( R1
" On )  <->  E. x  e.  On  A  e.  ( R1 `  x ) )
173, 13, 163imtr4i 258 . 2  |-  ( A 
C_  U. ( R1 " On )  ->  A  e. 
U. ( R1 " On ) )
181, 17impbii 181 1  |-  ( A  e.  U. ( R1
" On )  <->  A  C_  U. ( R1 " On ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   _Vcvv 2916    C_ wss 3280   U.cuni 3975   U_ciun 4053   Oncon0 4541   "cima 4840   Fun wfun 5407    Fn wfn 5408   ` cfv 5413   R1cr1 7644
This theorem is referenced by:  unir1  7695  tcwf  7763  tcrank  7764  rankcf  8608  wfgru  8647
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-recs 6592  df-rdg 6627  df-r1 7646
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