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Theorem r1elss 8215
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 8214 . 2  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  U. ( R1
" On ) )
2 r1elss.1 . . . 4  |-  A  e. 
_V
32tz9.12 8199 . . 3  |-  ( A. y  e.  A  E. x  e.  On  y  e.  ( R1 `  x
)  ->  E. x  e.  On  A  e.  ( R1 `  x ) )
4 dfss3 3489 . . . 4  |-  ( A 
C_  U. ( R1 " On )  <->  A. y  e.  A  y  e.  U. ( R1 " On ) )
5 r1fnon 8176 . . . . . . . 8  |-  R1  Fn  On
6 fnfun 5671 . . . . . . . 8  |-  ( R1  Fn  On  ->  Fun  R1 )
7 funiunfv 6141 . . . . . . . 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 2540 . . . . . 6  |-  ( y  e.  U_ x  e.  On  ( R1 `  x )  <->  y  e.  U. ( R1 " On ) )
10 eliun 4325 . . . . . 6  |-  ( y  e.  U_ x  e.  On  ( R1 `  x )  <->  E. x  e.  On  y  e.  ( R1 `  x ) )
119, 10bitr3i 251 . . . . 5  |-  ( y  e.  U. ( R1
" On )  <->  E. x  e.  On  y  e.  ( R1 `  x ) )
1211ralbii 2890 . . . 4  |-  ( A. y  e.  A  y  e.  U. ( R1 " On )  <->  A. y  e.  A  E. x  e.  On  y  e.  ( R1 `  x ) )
134, 12bitri 249 . . 3  |-  ( A 
C_  U. ( R1 " On )  <->  A. y  e.  A  E. x  e.  On  y  e.  ( R1 `  x ) )
148eleq2i 2540 . . . 4  |-  ( A  e.  U_ x  e.  On  ( R1 `  x )  <->  A  e.  U. ( R1 " On ) )
15 eliun 4325 . . . 4  |-  ( A  e.  U_ x  e.  On  ( R1 `  x )  <->  E. x  e.  On  A  e.  ( R1 `  x ) )
1614, 15bitr3i 251 . . 3  |-  ( A  e.  U. ( R1
" On )  <->  E. x  e.  On  A  e.  ( R1 `  x ) )
173, 13, 163imtr4i 266 . 2  |-  ( A 
C_  U. ( R1 " On )  ->  A  e. 
U. ( R1 " On ) )
181, 17impbii 188 1  |-  ( A  e.  U. ( R1
" On )  <->  A  C_  U. ( R1 " On ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1374    e. wcel 1762   A.wral 2809   E.wrex 2810   _Vcvv 3108    C_ wss 3471   U.cuni 4240   U_ciun 4320   Oncon0 4873   "cima 4997   Fun wfun 5575    Fn wfn 5576   ` cfv 5581   R1cr1 8171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-om 6674  df-recs 7034  df-rdg 7068  df-r1 8173
This theorem is referenced by:  unir1  8222  tcwf  8292  tcrank  8293  rankcf  9146  wfgru  9185
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