Theorem tcmin 9499

Description: Defining property of the transitive closure function: it is a subset of any transitive class containing 𝐴. (Contributed by Mario Carneiro, 23-Jun-2013.)

Assertion

Ref	Expression
tcmin	⊢ (𝐴 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → (TC‘𝐴) ⊆ 𝐵))

Proof of Theorem tcmin

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tcvalg 9496	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
2		fvex 6787	. . . . 5 ⊢ (TC‘𝐴) ∈ V
3	1, 2	eqeltrrdi 2848	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V)
4		intexab 5263	. . . 4 ⊢ (∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥) ↔ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V)
5	3, 4	sylibr 233	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
6		ssin 4164	. . . . . . . . 9 ⊢ ((𝐴 ⊆ 𝑥 ∧ 𝐴 ⊆ 𝐵) ↔ 𝐴 ⊆ (𝑥 ∩ 𝐵))
7	6	biimpi 215	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝑥 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ (𝑥 ∩ 𝐵))
8		trin 5201	. . . . . . . 8 ⊢ ((Tr 𝑥 ∧ Tr 𝐵) → Tr (𝑥 ∩ 𝐵))
9	7, 8	anim12i 613	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝑥 ∧ 𝐴 ⊆ 𝐵) ∧ (Tr 𝑥 ∧ Tr 𝐵)) → (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵)))
10	9	an4s 657	. . . . . 6 ⊢ (((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) ∧ (𝐴 ⊆ 𝐵 ∧ Tr 𝐵)) → (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵)))
11	10	expcom 414	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵))))
12		vex 3436	. . . . . . . . 9 ⊢ 𝑥 ∈ V
13	12	inex1 5241	. . . . . . . 8 ⊢ (𝑥 ∩ 𝐵) ∈ V
14		sseq2 3947	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → (𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ (𝑥 ∩ 𝐵)))
15		treq 5197	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → (Tr 𝑦 ↔ Tr (𝑥 ∩ 𝐵)))
16	14, 15	anbi12d 631	. . . . . . . 8 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) ↔ (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵))))
17	13, 16	elab 3609	. . . . . . 7 ⊢ ((𝑥 ∩ 𝐵) ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ↔ (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵)))
18		intss1 4894	. . . . . . 7 ⊢ ((𝑥 ∩ 𝐵) ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ (𝑥 ∩ 𝐵))
19	17, 18	sylbir 234	. . . . . 6 ⊢ ((𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵)) → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ (𝑥 ∩ 𝐵))
20		inss2 4163	. . . . . 6 ⊢ (𝑥 ∩ 𝐵) ⊆ 𝐵
21	19, 20	sstrdi 3933	. . . . 5 ⊢ ((𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵)) → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵)
22	11, 21	syl6 35	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵))
23	22	exlimdv 1936	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → (∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵))
24	5, 23	syl5com 31	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵))
25		tcvalg 9496	. . 3 ⊢ (𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)})
26	25	sseq1d 3952	. 2 ⊢ (𝐴 ∈ 𝑉 → ((TC‘𝐴) ⊆ 𝐵 ↔ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵))
27	24, 26	sylibrd 258	1 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → (TC‘𝐴) ⊆ 𝐵))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  ∩ cint 4879  Tr wtr 5191  ‘cfv 6433  TCctc 9494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-tc 9495

This theorem is referenced by:  tcidm  9504  tc0  9505  tcwf  9641  itunitc  10177  grur1  10576