Theorem tcvalg 9648

Description: Value of the transitive closure function. (The fact that this intersection exists is a non-trivial fact that depends on ax-inf 9550; see tz9.1 9641.) (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 6827	. . 3 ⊢ (𝑦 = 𝐴 → (TC‘𝑦) = (TC‘𝐴))
2		sseq1 3940	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑥))
3	2	anbi1d 637	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑦 ⊆ 𝑥 ∧ Tr 𝑥) ↔ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)))
4	3	abbidv 2805	. . . 4 ⊢ (𝑦 = 𝐴 → {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)} = {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
5	4	inteqd 4882	. . 3 ⊢ (𝑦 = 𝐴 → ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)} = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
6	1, 5	eqeq12d 2755	. 2 ⊢ (𝑦 = 𝐴 → ((TC‘𝑦) = ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)} ↔ (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}))
7		vex 3435	. . 3 ⊢ 𝑦 ∈ V
8	7	tz9.1c 9642	. . 3 ⊢ ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V
9		df-tc 9647	. . . 4 ⊢ TC = (𝑦 ∈ V ↦ ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)})
10	9	fvmpt2 6947	. . 3 ⊢ ((𝑦 ∈ V ∧ ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V) → (TC‘𝑦) = ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)})
11	7, 8, 10	mp2an 698	. 2 ⊢ (TC‘𝑦) = ∩ {𝑥 ∣ (𝑦 ⊆ 𝑥 ∧ Tr 𝑥)}
12	6, 11	vtoclg 3500	1 ⊢ (𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {cab 2717  Vcvv 3431   ⊆ wss 3883  ∩ cint 4877  Tr wtr 5179  ‘cfv 6485  TCctc 9646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678  ax-inf2 9553

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-tc 9647

This theorem is referenced by:  tcid  9649  tctr  9650  tcmin  9651  tc2  9652  tcsni  9653  tcss  9654  tcel  9655  tcrank  9799