Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  tcwf Structured version   Visualization version   GIF version

Theorem tcwf 8629
 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 8551 . . 3 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
2 dftr3 4684 . . . . 5 (Tr (𝑅1 “ On) ↔ ∀𝑥 (𝑅1 “ On)𝑥 (𝑅1 “ On))
3 r1elssi 8551 . . . . 5 (𝑥 (𝑅1 “ On) → 𝑥 (𝑅1 “ On))
42, 3mprgbir 2911 . . . 4 Tr (𝑅1 “ On)
5 tcmin 8500 . . . 4 (𝐴 (𝑅1 “ On) → ((𝐴 (𝑅1 “ On) ∧ Tr (𝑅1 “ On)) → (TC‘𝐴) ⊆ (𝑅1 “ On)))
64, 5mpan2i 709 . . 3 (𝐴 (𝑅1 “ On) → (𝐴 (𝑅1 “ On) → (TC‘𝐴) ⊆ (𝑅1 “ On)))
71, 6mpd 15 . 2 (𝐴 (𝑅1 “ On) → (TC‘𝐴) ⊆ (𝑅1 “ On))
8 fvex 6113 . . 3 (TC‘𝐴) ∈ V
98r1elss 8552 . 2 ((TC‘𝐴) ∈ (𝑅1 “ On) ↔ (TC‘𝐴) ⊆ (𝑅1 “ On))
107, 9sylibr 223 1 (𝐴 (𝑅1 “ On) → (TC‘𝐴) ∈ (𝑅1 “ On))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977   ⊆ wss 3540  ∪ cuni 4372  Tr wtr 4680   “ cima 5041  Oncon0 5640  ‘cfv 5804  TCctc 8495  𝑅1cr1 8508 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-tc 8496  df-r1 8510 This theorem is referenced by:  tcrank  8630
 Copyright terms: Public domain W3C validator