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Theorem tcvalg 8558
Description: Value of the transitive closure function. (The fact that this intersection exists is a non-trivial fact that depends on ax-inf 8479; see tz9.1 8549.) (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 6148 . . 3 (𝑦 = 𝐴 → (TC‘𝑦) = (TC‘𝐴))
2 sseq1 3605 . . . . . 6 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
32anbi1d 740 . . . . 5 (𝑦 = 𝐴 → ((𝑦𝑥 ∧ Tr 𝑥) ↔ (𝐴𝑥 ∧ Tr 𝑥)))
43abbidv 2738 . . . 4 (𝑦 = 𝐴 → {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
54inteqd 4445 . . 3 (𝑦 = 𝐴 {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
61, 5eqeq12d 2636 . 2 (𝑦 = 𝐴 → ((TC‘𝑦) = {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} ↔ (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}))
7 vex 3189 . . 3 𝑦 ∈ V
87tz9.1c 8550 . . 3 {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} ∈ V
9 df-tc 8557 . . . 4 TC = (𝑦 ∈ V ↦ {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)})
109fvmpt2 6248 . . 3 ((𝑦 ∈ V ∧ {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} ∈ V) → (TC‘𝑦) = {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)})
117, 8, 10mp2an 707 . 2 (TC‘𝑦) = {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)}
126, 11vtoclg 3252 1 (𝐴𝑉 → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {cab 2607  Vcvv 3186  wss 3555   cint 4440  Tr wtr 4712  cfv 5847  TCctc 8556
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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482
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 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-tc 8557
This theorem is referenced by:  tcid  8559  tctr  8560  tcmin  8561  tc2  8562  tcsni  8563  tcss  8564  tcel  8565  tcrank  8691
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