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Theorem tctr 9780
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 5277 . . . 4 (∀𝑦 ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}Tr 𝑦 → Tr {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
2 vex 3484 . . . . . 6 𝑦 ∈ V
3 sseq2 4010 . . . . . . 7 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
4 treq 5267 . . . . . . 7 (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦))
53, 4anbi12d 632 . . . . . 6 (𝑥 = 𝑦 → ((𝐴𝑥 ∧ Tr 𝑥) ↔ (𝐴𝑦 ∧ Tr 𝑦)))
62, 5elab 3679 . . . . 5 (𝑦 ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ↔ (𝐴𝑦 ∧ Tr 𝑦))
76simprbi 496 . . . 4 (𝑦 ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} → Tr 𝑦)
81, 7mprg 3067 . . 3 Tr {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
9 tcvalg 9778 . . . 4 (𝐴 ∈ V → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
10 treq 5267 . . . 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 258 . 2 (𝐴 ∈ V → Tr (TC‘𝐴))
13 tr0 5272 . . 3 Tr ∅
14 fvprc 6898 . . . 4 𝐴 ∈ V → (TC‘𝐴) = ∅)
15 treq 5267 . . . 4 ((TC‘𝐴) = ∅ → (Tr (TC‘𝐴) ↔ Tr ∅))
1614, 15syl 17 . . 3 𝐴 ∈ V → (Tr (TC‘𝐴) ↔ Tr ∅))
1713, 16mpbiri 258 . 2 𝐴 ∈ V → Tr (TC‘𝐴))
1812, 17pm2.61i 182 1 Tr (TC‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  Vcvv 3480  wss 3951  c0 4333   cint 4946  Tr wtr 5259  cfv 6561  TCctc 9776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-tc 9777
This theorem is referenced by:  tc2  9782  tcidm  9786  itunitc1  10460  tcfr  44980
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