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Theorem tcwf 8706
Description: The transitive closure function is well-founded if its argument is. (Contributed by Mario Carneiro, 23-Jun-2013.)
Assertion
Ref Expression
tcwf (𝐴 (𝑅1 “ On) → (TC‘𝐴) ∈ (𝑅1 “ On))

Proof of Theorem tcwf
StepHypRef Expression
1 r1elssi 8628 . . 3 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
2 dftr3 4726 . . . . 5 (Tr (𝑅1 “ On) ↔ ∀𝑥 (𝑅1 “ On)𝑥 (𝑅1 “ On))
3 r1elssi 8628 . . . . 5 (𝑥 (𝑅1 “ On) → 𝑥 (𝑅1 “ On))
42, 3mprgbir 2923 . . . 4 Tr (𝑅1 “ On)
5 tcmin 8577 . . . 4 (𝐴 (𝑅1 “ On) → ((𝐴 (𝑅1 “ On) ∧ Tr (𝑅1 “ On)) → (TC‘𝐴) ⊆ (𝑅1 “ On)))
64, 5mpan2i 712 . . 3 (𝐴 (𝑅1 “ On) → (𝐴 (𝑅1 “ On) → (TC‘𝐴) ⊆ (𝑅1 “ On)))
71, 6mpd 15 . 2 (𝐴 (𝑅1 “ On) → (TC‘𝐴) ⊆ (𝑅1 “ On))
8 fvex 6168 . . 3 (TC‘𝐴) ∈ V
98r1elss 8629 . 2 ((TC‘𝐴) ∈ (𝑅1 “ On) ↔ (TC‘𝐴) ⊆ (𝑅1 “ On))
107, 9sylibr 224 1 (𝐴 (𝑅1 “ On) → (TC‘𝐴) ∈ (𝑅1 “ On))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  wss 3560   cuni 4409  Tr wtr 4722  cima 5087  Oncon0 5692  cfv 5857  TCctc 8572  𝑅1cr1 8585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-om 7028  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-tc 8573  df-r1 8587
This theorem is referenced by:  tcrank  8707
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