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Theorem tcwf 7553
Description: The transitive closure function is well-founded if its argument is. (Contributed by Mario Carneiro, 23-Jun-2013.)
Assertion
Ref Expression
tcwf  |-  ( A  e.  U. ( R1
" On )  -> 
( TC `  A
)  e.  U. ( R1 " On ) )

Proof of Theorem tcwf
StepHypRef Expression
1 r1elssi 7477 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  U. ( R1
" On ) )
2 dftr3 4117 . . . . 5  |-  ( Tr 
U. ( R1 " On )  <->  A. x  e.  U. ( R1 " On ) x  C_  U. ( R1 " On ) )
3 r1elssi 7477 . . . . 5  |-  ( x  e.  U. ( R1
" On )  ->  x  C_  U. ( R1
" On ) )
42, 3mprgbir 2613 . . . 4  |-  Tr  U. ( R1 " On )
5 tcmin 7426 . . . 4  |-  ( A  e.  U. ( R1
" On )  -> 
( ( A  C_  U. ( R1 " On )  /\  Tr  U. ( R1 " On ) )  ->  ( TC `  A )  C_  U. ( R1 " On ) ) )
64, 5mpan2i 658 . . 3  |-  ( A  e.  U. ( R1
" On )  -> 
( A  C_  U. ( R1 " On )  -> 
( TC `  A
)  C_  U. ( R1 " On ) ) )
71, 6mpd 14 . 2  |-  ( A  e.  U. ( R1
" On )  -> 
( TC `  A
)  C_  U. ( R1 " On ) )
8 fvex 5539 . . 3  |-  ( TC
`  A )  e. 
_V
98r1elss 7478 . 2  |-  ( ( TC `  A )  e.  U. ( R1
" On )  <->  ( TC `  A )  C_  U. ( R1 " On ) )
107, 9sylibr 203 1  |-  ( A  e.  U. ( R1
" On )  -> 
( TC `  A
)  e.  U. ( R1 " On ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684    C_ wss 3152   U.cuni 3827   Tr wtr 4113   Oncon0 4392   "cima 4692   ` cfv 5255   TCctc 7421   R1cr1 7434
This theorem is referenced by:  tcrank  7554
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
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-int 3863  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-tc 7422  df-r1 7436
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