Theorem tcvalg 9621

Description: Value of the transitive closure function. (The fact that this intersection exists is a non-trivial fact that depends on ax-inf 9523; see tz9.1 9614.) (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 6817	. . 3 ⊢ (𝑦 = 𝐴 → (TC‘𝑦) = (TC‘𝐴))
2		sseq1 3955	. . . . . 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 2797	. . . 4 ⊢ (𝑦 = 𝐴 → {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)} = {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
5	4	inteqd 4897	. . 3 ⊢ (𝑦 = 𝐴 → ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)} = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
6	1, 5	eqeq12d 2747	. 2 ⊢ (𝑦 = 𝐴 → ((TC‘𝑦) = ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)} ↔ (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}))
7		vex 3440	. . 3 ⊢ 𝑦 ∈ V
8	7	tz9.1c 9615	. . 3 ⊢ ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V
9		df-tc 9620	. . . 4 ⊢ TC = (𝑦 ∈ V ↦ ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)})
10	9	fvmpt2 6935	. . 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 3507	1 ⊢ (𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  Vcvv 3436   ⊆ wss 3897  ∩ cint 4892  Tr wtr 5193  ‘cfv 6476  TCctc 9619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663  ax-inf2 9526

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-om 7792  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-tc 9620

This theorem is referenced by:  tcid  9622  tctr  9623  tcmin  9624  tc2  9625  tcsni  9626  tcss  9627  tcel  9628  tcrank  9772