Theorem tcel 9655

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 9648	. 2 ⊢ (𝐵 ∈ 𝐴 → (TC‘𝐵) = ∩ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)})
2		ssel 3916	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝑥 → (𝐵 ∈ 𝐴 → 𝐵 ∈ 𝑥))
3		trss 5203	. . . . . . . . 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 4006	. . . 4 ⊢ (𝐵 ∈ 𝐴 → {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)})
10		intss 4912	. . . 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 9648	. . . 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 3964	. 2 ⊢ (𝐵 ∈ 𝐴 → ∩ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ (TC‘𝐴))
16	1, 15	eqsstrd 3957	1 ⊢ (𝐵 ∈ 𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  Vcvv 3430   ⊆ wss 3890  ∩ cint 4890  Tr wtr 5193  ‘cfv 6492  TCctc 9646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682  ax-inf2 9553

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-tc 9647

This theorem is referenced by:  tcrank  9799  hsmexlem4  10342