Theorem tcss 9502

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

Hypothesis

Ref	Expression
tc2.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
tcss	⊢ (𝐵 ⊆ 𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))

Proof of Theorem tcss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tc2.1	. . . 4 ⊢ 𝐴 ∈ V
2	1	ssex 5245	. . 3 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 ∈ V)
3		tcvalg 9496	. . 3 ⊢ (𝐵 ∈ V → (TC‘𝐵) = ∩ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)})
4	2, 3	syl 17	. 2 ⊢ (𝐵 ⊆ 𝐴 → (TC‘𝐵) = ∩ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)})
5		sstr2 3928	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ⊆ 𝑥 → 𝐵 ⊆ 𝑥))
6	5	anim1d 611	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)))
7	6	ss2abdv 3997	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)})
8		intss 4900	. . . 4 ⊢ ({𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)} → ∩ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
9	7, 8	syl 17	. . 3 ⊢ (𝐵 ⊆ 𝐴 → ∩ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
10		tcvalg 9496	. . . 4 ⊢ (𝐴 ∈ V → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
11	1, 10	ax-mp 5	. . 3 ⊢ (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}
12	9, 11	sseqtrrdi 3972	. 2 ⊢ (𝐵 ⊆ 𝐴 → ∩ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ (TC‘𝐴))
13	4, 12	eqsstrd 3959	1 ⊢ (𝐵 ⊆ 𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {cab 2715  Vcvv 3432   ⊆ 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:  hsmexlem4  10185