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Theorem tcvalg 8776
Description: Value of the transitive closure function. (The fact that this intersection exists is a non-trivial fact that depends on ax-inf 8697; see tz9.1 8767.) (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 6330 . . 3 (𝑦 = 𝐴 → (TC‘𝑦) = (TC‘𝐴))
2 sseq1 3775 . . . . . 6 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
32anbi1d 615 . . . . 5 (𝑦 = 𝐴 → ((𝑦𝑥 ∧ Tr 𝑥) ↔ (𝐴𝑥 ∧ Tr 𝑥)))
43abbidv 2890 . . . 4 (𝑦 = 𝐴 → {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
54inteqd 4616 . . 3 (𝑦 = 𝐴 {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
61, 5eqeq12d 2786 . 2 (𝑦 = 𝐴 → ((TC‘𝑦) = {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} ↔ (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}))
7 vex 3354 . . 3 𝑦 ∈ V
87tz9.1c 8768 . . 3 {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} ∈ V
9 df-tc 8775 . . . 4 TC = (𝑦 ∈ V ↦ {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)})
109fvmpt2 6431 . . 3 ((𝑦 ∈ V ∧ {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} ∈ V) → (TC‘𝑦) = {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)})
117, 8, 10mp2an 672 . 2 (TC‘𝑦) = {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)}
126, 11vtoclg 3417 1 (𝐴𝑉 → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  {cab 2757  Vcvv 3351  wss 3723   cint 4611  Tr wtr 4886  cfv 6029  TCctc 8774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094  ax-inf2 8700
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-om 7211  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-tc 8775
This theorem is referenced by:  tcid  8777  tctr  8778  tcmin  8779  tc2  8780  tcsni  8781  tcss  8782  tcel  8783  tcrank  8909
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