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Theorem tctr 9778
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 5283 . . . 4 (∀𝑦 ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}Tr 𝑦 → Tr {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
2 vex 3482 . . . . . 6 𝑦 ∈ V
3 sseq2 4022 . . . . . . 7 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
4 treq 5273 . . . . . . 7 (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦))
53, 4anbi12d 632 . . . . . 6 (𝑥 = 𝑦 → ((𝐴𝑥 ∧ Tr 𝑥) ↔ (𝐴𝑦 ∧ Tr 𝑦)))
62, 5elab 3681 . . . . 5 (𝑦 ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ↔ (𝐴𝑦 ∧ Tr 𝑦))
76simprbi 496 . . . 4 (𝑦 ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} → Tr 𝑦)
81, 7mprg 3065 . . 3 Tr {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
9 tcvalg 9776 . . . 4 (𝐴 ∈ V → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
10 treq 5273 . . . 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 5278 . . 3 Tr ∅
14 fvprc 6899 . . . 4 𝐴 ∈ V → (TC‘𝐴) = ∅)
15 treq 5273 . . . 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 1537  wcel 2106  {cab 2712  Vcvv 3478  wss 3963  c0 4339   cint 4951  Tr wtr 5265  cfv 6563  TCctc 9774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-tc 9775
This theorem is referenced by:  tc2  9780  tcidm  9784  itunitc1  10458
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