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Theorem tcvalg 8497
 Description: Value of the transitive closure function. (The fact that this intersection exists is a non-trivial fact that depends on ax-inf 8418; see tz9.1 8488.) (Contributed by Mario Carneiro, 23-Jun-2013.)
Assertion
Ref Expression
tcvalg (𝐴𝑉 → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem tcvalg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6103 . . 3 (𝑦 = 𝐴 → (TC‘𝑦) = (TC‘𝐴))
2 sseq1 3589 . . . . . 6 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
32anbi1d 737 . . . . 5 (𝑦 = 𝐴 → ((𝑦𝑥 ∧ Tr 𝑥) ↔ (𝐴𝑥 ∧ Tr 𝑥)))
43abbidv 2728 . . . 4 (𝑦 = 𝐴 → {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
54inteqd 4415 . . 3 (𝑦 = 𝐴 {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
61, 5eqeq12d 2625 . 2 (𝑦 = 𝐴 → ((TC‘𝑦) = {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} ↔ (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}))
7 vex 3176 . . 3 𝑦 ∈ V
87tz9.1c 8489 . . 3 {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} ∈ V
9 df-tc 8496 . . . 4 TC = (𝑦 ∈ V ↦ {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)})
109fvmpt2 6200 . . 3 ((𝑦 ∈ V ∧ {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} ∈ V) → (TC‘𝑦) = {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)})
117, 8, 10mp2an 704 . 2 (TC‘𝑦) = {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)}
126, 11vtoclg 3239 1 (𝐴𝑉 → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173   ⊆ wss 3540  ∩ cint 4410  Tr wtr 4680  ‘cfv 5804  TCctc 8495 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 This theorem is referenced by:  tcid  8498  tctr  8499  tcmin  8500  tc2  8501  tcsni  8502  tcss  8503  tcel  8504  tcrank  8630
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