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Theorem tcvalg 9172
Description: Value of the transitive closure function. (The fact that this intersection exists is a non-trivial fact that depends on ax-inf 9093; see tz9.1 9163.) (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 6666 . . 3 (𝑦 = 𝐴 → (TC‘𝑦) = (TC‘𝐴))
2 sseq1 3995 . . . . . 6 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
32anbi1d 629 . . . . 5 (𝑦 = 𝐴 → ((𝑦𝑥 ∧ Tr 𝑥) ↔ (𝐴𝑥 ∧ Tr 𝑥)))
43abbidv 2889 . . . 4 (𝑦 = 𝐴 → {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
54inteqd 4878 . . 3 (𝑦 = 𝐴 {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
61, 5eqeq12d 2841 . 2 (𝑦 = 𝐴 → ((TC‘𝑦) = {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} ↔ (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}))
7 vex 3502 . . 3 𝑦 ∈ V
87tz9.1c 9164 . . 3 {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} ∈ V
9 df-tc 9171 . . . 4 TC = (𝑦 ∈ V ↦ {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)})
109fvmpt2 6774 . . 3 ((𝑦 ∈ V ∧ {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} ∈ V) → (TC‘𝑦) = {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)})
117, 8, 10mp2an 688 . 2 (TC‘𝑦) = {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)}
126, 11vtoclg 3572 1 (𝐴𝑉 → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  {cab 2803  Vcvv 3499  wss 3939   cint 4873  Tr wtr 5168  cfv 6351  TCctc 9170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-inf2 9096
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-om 7572  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-tc 9171
This theorem is referenced by:  tcid  9173  tctr  9174  tcmin  9175  tc2  9176  tcsni  9177  tcss  9178  tcel  9179  tcrank  9305
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