Theorem tcel 8794

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 8787	. 2 ⊢ (𝐵 ∈ 𝐴 → (TC‘𝐵) = ∩ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)})
2		ssel 3738	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝑥 → (𝐵 ∈ 𝐴 → 𝐵 ∈ 𝑥))
3		trss 4913	. . . . . . . . 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 446	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → 𝐵 ⊆ 𝑥))
7		simpr 479	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → Tr 𝑥)
8	7	a1i 11	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → Tr 𝑥))
9	6, 8	jcad 556	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)))
10	9	ss2abdv 3816	. . . 4 ⊢ (𝐵 ∈ 𝐴 → {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)})
11		intss 4650	. . . 4 ⊢ ({𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)} → ∩ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
12	10, 11	syl 17	. . 3 ⊢ (𝐵 ∈ 𝐴 → ∩ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
13		tc2.1	. . . 4 ⊢ 𝐴 ∈ V
14		tcvalg 8787	. . . 4 ⊢ (𝐴 ∈ V → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
15	13, 14	ax-mp 5	. . 3 ⊢ (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}
16	12, 15	syl6sseqr 3793	. 2 ⊢ (𝐵 ∈ 𝐴 → ∩ {𝑥 ∣ (𝐵 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ (TC‘𝐴))
17	1, 16	eqsstrd 3780	1 ⊢ (𝐵 ∈ 𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  {cab 2746  Vcvv 3340   ⊆ wss 3715  ∩ cint 4627  Tr wtr 4904  ‘cfv 6049  TCctc 8785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-tc 8786

This theorem is referenced by:  tcrank  8920  hsmexlem4  9443