Theorem tcvalg 9691

Description: Value of the transitive closure function. (The fact that this intersection exists is a non-trivial fact that depends on ax-inf 9591; see tz9.1 9682.) (Contributed by Mario Carneiro, 23-Jun-2013.)

Assertion

Ref	Expression
tcvalg	⊢ (𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 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.

Step	Hyp	Ref	Expression
1		fveq2 6858	. . 3 ⊢ (𝑦 = 𝐴 → (TC‘𝑦) = (TC‘𝐴))
2		sseq1 3972	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑥))
3	2	anbi1d 631	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑦 ⊆ 𝑥 ∧ Tr 𝑥) ↔ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)))
4	3	abbidv 2795	. . . 4 ⊢ (𝑦 = 𝐴 → {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)} = {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
5	4	inteqd 4915	. . 3 ⊢ (𝑦 = 𝐴 → ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)} = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
6	1, 5	eqeq12d 2745	. 2 ⊢ (𝑦 = 𝐴 → ((TC‘𝑦) = ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)} ↔ (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}))
7		vex 3451	. . 3 ⊢ 𝑦 ∈ V
8	7	tz9.1c 9683	. . 3 ⊢ ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V
9		df-tc 9690	. . . 4 ⊢ TC = (𝑦 ∈ V ↦ ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)})
10	9	fvmpt2 6979	. . 3 ⊢ ((𝑦 ∈ V ∧ ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V) → (TC‘𝑦) = ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)})
11	7, 8, 10	mp2an 692	. 2 ⊢ (TC‘𝑦) = ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)}
12	6, 11	vtoclg 3520	1 ⊢ (𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  Vcvv 3447   ⊆ wss 3914  ∩ cint 4910  Tr wtr 5214  ‘cfv 6511  TCctc 9689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711  ax-inf2 9594

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-tc 9690

This theorem is referenced by:  tcid  9692  tctr  9693  tcmin  9694  tc2  9695  tcsni  9696  tcss  9697  tcel  9698  tcrank  9837