Theorem tcel 9785

Description: The transitive closure function converts the element relation to the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.)

Hypothesis

Ref	Expression
tc2.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
tcel	⊢ (𝐵 ∈ 𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))

Proof of Theorem tcel

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tcvalg 9778	. 2 ⊢ (𝐵 ∈ 𝐴 → (TC‘𝐵) = ∩ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)})
2		ssel 3977	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝑥 → (𝐵 ∈ 𝐴 → 𝐵 ∈ 𝑥))
3		trss 5270	. . . . . . . . 9 ⊢ (Tr 𝑥 → (𝐵 ∈ 𝑥 → 𝐵 ⊆ 𝑥))
4	3	com12 32	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑥 → (Tr 𝑥 → 𝐵 ⊆ 𝑥))
5	2, 4	syl6com 37	. . . . . . 7 ⊢ (𝐵 ∈ 𝐴 → (𝐴 ⊆ 𝑥 → (Tr 𝑥 → 𝐵 ⊆ 𝑥)))
6	5	impd 410	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → 𝐵 ⊆ 𝑥))
7		simpr 484	. . . . . 6 ⊢ ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → Tr 𝑥)
8	6, 7	jca2 513	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)))
9	8	ss2abdv 4066	. . . 4 ⊢ (𝐵 ∈ 𝐴 → {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)})
10		intss 4969	. . . 4 ⊢ ({𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)} → ∩ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
11	9, 10	syl 17	. . 3 ⊢ (𝐵 ∈ 𝐴 → ∩ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
12		tc2.1	. . . 4 ⊢ 𝐴 ∈ V
13		tcvalg 9778	. . . 4 ⊢ (𝐴 ∈ V → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
14	12, 13	ax-mp 5	. . 3 ⊢ (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}
15	11, 14	sseqtrrdi 4025	. 2 ⊢ (𝐵 ∈ 𝐴 → ∩ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ (TC‘𝐴))
16	1, 15	eqsstrd 4018	1 ⊢ (𝐵 ∈ 𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2714  Vcvv 3480   ⊆ wss 3951  ∩ cint 4946  Tr wtr 5259  ‘cfv 6561  TCctc 9776

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-inf2 9681

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-tc 9777

This theorem is referenced by:  tcrank  9924  hsmexlem4  10469