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Theorem tcwf 7741
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 7665 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  U. ( R1
" On ) )
2 dftr3 4248 . . . . 5  |-  ( Tr 
U. ( R1 " On )  <->  A. x  e.  U. ( R1 " On ) x  C_  U. ( R1 " On ) )
3 r1elssi 7665 . . . . 5  |-  ( x  e.  U. ( R1
" On )  ->  x  C_  U. ( R1
" On ) )
42, 3mprgbir 2720 . . . 4  |-  Tr  U. ( R1 " On )
5 tcmin 7614 . . . 4  |-  ( A  e.  U. ( R1
" On )  -> 
( ( A  C_  U. ( R1 " On )  /\  Tr  U. ( R1 " On ) )  ->  ( TC `  A )  C_  U. ( R1 " On ) ) )
64, 5mpan2i 659 . . 3  |-  ( A  e.  U. ( R1
" On )  -> 
( A  C_  U. ( R1 " On )  -> 
( TC `  A
)  C_  U. ( R1 " On ) ) )
71, 6mpd 15 . 2  |-  ( A  e.  U. ( R1
" On )  -> 
( TC `  A
)  C_  U. ( R1 " On ) )
8 fvex 5683 . . 3  |-  ( TC
`  A )  e. 
_V
98r1elss 7666 . 2  |-  ( ( TC `  A )  e.  U. ( R1
" On )  <->  ( TC `  A )  C_  U. ( R1 " On ) )
107, 9sylibr 204 1  |-  ( A  e.  U. ( R1
" On )  -> 
( TC `  A
)  e.  U. ( R1 " On ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717    C_ wss 3264   U.cuni 3958   Tr wtr 4244   Oncon0 4523   "cima 4822   ` cfv 5395   TCctc 7609   R1cr1 7622
This theorem is referenced by:  tcrank  7742
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-recs 6570  df-rdg 6605  df-tc 7610  df-r1 7624
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