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Theorem tcwf 7799
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 7723 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  U. ( R1
" On ) )
2 dftr3 4298 . . . . 5  |-  ( Tr 
U. ( R1 " On )  <->  A. x  e.  U. ( R1 " On ) x  C_  U. ( R1 " On ) )
3 r1elssi 7723 . . . . 5  |-  ( x  e.  U. ( R1
" On )  ->  x  C_  U. ( R1
" On ) )
42, 3mprgbir 2768 . . . 4  |-  Tr  U. ( R1 " On )
5 tcmin 7672 . . . 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 5734 . . 3  |-  ( TC
`  A )  e. 
_V
98r1elss 7724 . 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 1725    C_ wss 3312   U.cuni 4007   Tr wtr 4294   Oncon0 4573   "cima 4873   ` cfv 5446   TCctc 7667   R1cr1 7680
This theorem is referenced by:  tcrank  7800
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-recs 6625  df-rdg 6660  df-tc 7668  df-r1 7682
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