Theorem tcel 9783

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 9776	. 2 ⊢ (𝐵 ∈ 𝐴 → (TC‘𝐵) = ∩ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)})
2		ssel 3989	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝑥 → (𝐵 ∈ 𝐴 → 𝐵 ∈ 𝑥))
3		trss 5276	. . . . . . . . 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 4076	. . . 4 ⊢ (𝐵 ∈ 𝐴 → {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)})
10		intss 4974	. . . 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 9776	. . . 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 4047	. 2 ⊢ (𝐵 ∈ 𝐴 → ∩ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ (TC‘𝐴))
16	1, 15	eqsstrd 4034	1 ⊢ (𝐵 ∈ 𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {cab 2712  Vcvv 3478   ⊆ wss 3963  ∩ cint 4951  Tr wtr 5265  ‘cfv 6563  TCctc 9774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-inf2 9679

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-tc 9775

This theorem is referenced by:  tcrank  9922  hsmexlem4  10467