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Theorem tcvalg 9674
Description: Value of the transitive closure function. (The fact that this intersection exists is a non-trivial fact that depends on ax-inf 9574; see tz9.1 9665.) (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 6842 . . 3 (𝑦 = 𝐴 → (TC‘𝑦) = (TC‘𝐴))
2 sseq1 3969 . . . . . 6 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
32anbi1d 630 . . . . 5 (𝑦 = 𝐴 → ((𝑦𝑥 ∧ Tr 𝑥) ↔ (𝐴𝑥 ∧ Tr 𝑥)))
43abbidv 2805 . . . 4 (𝑦 = 𝐴 → {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
54inteqd 4912 . . 3 (𝑦 = 𝐴 {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
61, 5eqeq12d 2752 . 2 (𝑦 = 𝐴 → ((TC‘𝑦) = {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} ↔ (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}))
7 vex 3449 . . 3 𝑦 ∈ V
87tz9.1c 9666 . . 3 {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} ∈ V
9 df-tc 9673 . . . 4 TC = (𝑦 ∈ V ↦ {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)})
109fvmpt2 6959 . . 3 ((𝑦 ∈ V ∧ {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)} ∈ V) → (TC‘𝑦) = {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)})
117, 8, 10mp2an 690 . 2 (TC‘𝑦) = {𝑥 ∣ (𝑦𝑥 ∧ Tr 𝑥)}
126, 11vtoclg 3525 1 (𝐴𝑉 → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {cab 2713  Vcvv 3445  wss 3910   cint 4907  Tr wtr 5222  cfv 6496  TCctc 9672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672  ax-inf2 9577
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-tc 9673
This theorem is referenced by:  tcid  9675  tctr  9676  tcmin  9677  tc2  9678  tcsni  9679  tcss  9680  tcel  9681  tcrank  9820
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