Theorem tcvalg 9744

Description: Value of the transitive closure function. (The fact that this intersection exists is a non-trivial fact that depends on ax-inf 9646; see tz9.1 9737.) (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 6890	. . 3 ⊢ (𝑦 = 𝐴 → (TC‘𝑦) = (TC‘𝐴))
2		sseq1 3934	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑥))
3	2	anbi1d 643	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑦 ⊆ 𝑥 ∧ Tr 𝑥) ↔ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)))
4	3	abbidv 2826	. . . 4 ⊢ (𝑦 = 𝐴 → {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)} = {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
5	4	inteqd 4911	. . 3 ⊢ (𝑦 = 𝐴 → ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)} = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
6	1, 5	eqeq12d 2776	. 2 ⊢ (𝑦 = 𝐴 → ((TC‘𝑦) = ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)} ↔ (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}))
7		vex 3454	. . 3 ⊢ 𝑦 ∈ V
8	7	tz9.1c 9738	. . 3 ⊢ ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V
9		df-tc 9743	. . . 4 ⊢ TC = (𝑦 ∈ V ↦ ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)})
10	9	fvmpt2 7010	. . 3 ⊢ ((𝑦 ∈ V ∧ ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V) → (TC‘𝑦) = ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)})
11	7, 8, 10	mp2an 705	. 2 ⊢ (TC‘𝑦) = ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)}
12	6, 11	vtoclg 3517	1 ⊢ (𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  Vcvv 3450   ⊆ wss 3896  ∩ cint 4906  Tr wtr 5211  ‘cfv 6543  TCctc 9742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5391  ax-un 7755  ax-inf2 9649

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-om 7882  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-tc 9743

This theorem is used by:  tcid  9745  tctr  9746  tcmin  9747  tc2  9748  tcsni  9749  tcss  9750  tcel  9751  tcrank  9908