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Theorem tctr 8576
Description: Defining property of the transitive closure function: it is transitive. (Contributed by Mario Carneiro, 23-Jun-2013.)
Assertion
Ref Expression
tctr Tr (TC‘𝐴)

Proof of Theorem tctr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 trint 4738 . . . 4 (∀𝑦 ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}Tr 𝑦 → Tr {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
2 vex 3193 . . . . . 6 𝑦 ∈ V
3 sseq2 3612 . . . . . . 7 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
4 treq 4728 . . . . . . 7 (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦))
53, 4anbi12d 746 . . . . . 6 (𝑥 = 𝑦 → ((𝐴𝑥 ∧ Tr 𝑥) ↔ (𝐴𝑦 ∧ Tr 𝑦)))
62, 5elab 3338 . . . . 5 (𝑦 ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ↔ (𝐴𝑦 ∧ Tr 𝑦))
76simprbi 480 . . . 4 (𝑦 ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} → Tr 𝑦)
81, 7mprg 2922 . . 3 Tr {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
9 tcvalg 8574 . . . 4 (𝐴 ∈ V → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
10 treq 4728 . . . 4 ((TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} → (Tr (TC‘𝐴) ↔ Tr {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}))
119, 10syl 17 . . 3 (𝐴 ∈ V → (Tr (TC‘𝐴) ↔ Tr {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}))
128, 11mpbiri 248 . 2 (𝐴 ∈ V → Tr (TC‘𝐴))
13 tr0 4734 . . 3 Tr ∅
14 fvprc 6152 . . . 4 𝐴 ∈ V → (TC‘𝐴) = ∅)
15 treq 4728 . . . 4 ((TC‘𝐴) = ∅ → (Tr (TC‘𝐴) ↔ Tr ∅))
1614, 15syl 17 . . 3 𝐴 ∈ V → (Tr (TC‘𝐴) ↔ Tr ∅))
1713, 16mpbiri 248 . 2 𝐴 ∈ V → Tr (TC‘𝐴))
1812, 17pm2.61i 176 1 Tr (TC‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 384   = wceq 1480  wcel 1987  {cab 2607  Vcvv 3190  wss 3560  c0 3897   cint 4447  Tr wtr 4722  cfv 5857  TCctc 8572
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
This theorem is referenced by:  tc2  8578  tcidm  8582  itunitc1  9202
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