Theorem tcvalg 9164

Description: Value of the transitive closure function. (The fact that this intersection exists is a non-trivial fact that depends on ax-inf 9085; see tz9.1 9155.) (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 6645	. . 3 ⊢ (𝑦 = 𝐴 → (TC‘𝑦) = (TC‘𝐴))
2		sseq1 3940	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑥))
3	2	anbi1d 632	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑦 ⊆ 𝑥 ∧ Tr 𝑥) ↔ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)))
4	3	abbidv 2862	. . . 4 ⊢ (𝑦 = 𝐴 → {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)} = {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
5	4	inteqd 4843	. . 3 ⊢ (𝑦 = 𝐴 → ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)} = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
6	1, 5	eqeq12d 2814	. 2 ⊢ (𝑦 = 𝐴 → ((TC‘𝑦) = ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)} ↔ (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}))
7		vex 3444	. . 3 ⊢ 𝑦 ∈ V
8	7	tz9.1c 9156	. . 3 ⊢ ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V
9		df-tc 9163	. . . 4 ⊢ TC = (𝑦 ∈ V ↦ ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)})
10	9	fvmpt2 6756	. . 3 ⊢ ((𝑦 ∈ V ∧ ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V) → (TC‘𝑦) = ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)})
11	7, 8, 10	mp2an 691	. 2 ⊢ (TC‘𝑦) = ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)}
12	6, 11	vtoclg 3515	1 ⊢ (𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776  Vcvv 3441   ⊆ wss 3881  ∩ cint 4838  Tr wtr 5136  ‘cfv 6324  TCctc 9162

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-tc 9163

This theorem is referenced by:  tcid  9165  tctr  9166  tcmin  9167  tc2  9168  tcsni  9169  tcss  9170  tcel  9171  tcrank  9297