Theorem tc2 9695

Description: A variant of the definition of the transitive closure function, using instead the smallest transitive set containing 𝐴 as a member, gives almost the same set, except that 𝐴 itself must be added because it is not usually a member of (TC‘𝐴) (and it is never a member if 𝐴 is well-founded). (Contributed by Mario Carneiro, 23-Jun-2013.)

Hypothesis

Ref	Expression
tc2.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
tc2	⊢ ((TC‘𝐴) ∪ {𝐴}) = ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}

Distinct variable group:   𝑥,𝐴

Proof of Theorem tc2

Step	Hyp	Ref	Expression
1		tc2.1	. . . . 5 ⊢ 𝐴 ∈ V
2		tcvalg 9691	. . . . 5 ⊢ (𝐴 ∈ V → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
3	1, 2	ax-mp 5	. . . 4 ⊢ (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}
4		trss 5225	. . . . . . 7 ⊢ (Tr 𝑥 → (𝐴 ∈ 𝑥 → 𝐴 ⊆ 𝑥))
5	4	imdistanri 569	. . . . . 6 ⊢ ((𝐴 ∈ 𝑥 ∧ Tr 𝑥) → (𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
6	5	ss2abi 4030	. . . . 5 ⊢ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}
7		intss 4933	. . . . 5 ⊢ ({𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)})
8	6, 7	ax-mp 5	. . . 4 ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}
9	3, 8	eqsstri 3993	. . 3 ⊢ (TC‘𝐴) ⊆ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}
10	1	elintab 4922	. . . . 5 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)} ↔ ∀𝑥((𝐴 ∈ 𝑥 ∧ Tr 𝑥) → 𝐴 ∈ 𝑥))
11		simpl 482	. . . . 5 ⊢ ((𝐴 ∈ 𝑥 ∧ Tr 𝑥) → 𝐴 ∈ 𝑥)
12	10, 11	mpgbir 1799	. . . 4 ⊢ 𝐴 ∈ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}
13	1	snss 4749	. . . 4 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)} ↔ {𝐴} ⊆ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)})
14	12, 13	mpbi 230	. . 3 ⊢ {𝐴} ⊆ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}
15	9, 14	unssi 4154	. 2 ⊢ ((TC‘𝐴) ∪ {𝐴}) ⊆ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}
16	1	snid 4626	. . . . 5 ⊢ 𝐴 ∈ {𝐴}
17		elun2 4146	. . . . 5 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}))
18	16, 17	ax-mp 5	. . . 4 ⊢ 𝐴 ∈ ((TC‘𝐴) ∪ {𝐴})
19		uniun 4894	. . . . . . 7 ⊢ ∪ ((TC‘𝐴) ∪ {𝐴}) = (∪ (TC‘𝐴) ∪ ∪ {𝐴})
20		tctr 9693	. . . . . . . . 9 ⊢ Tr (TC‘𝐴)
21		df-tr 5215	. . . . . . . . 9 ⊢ (Tr (TC‘𝐴) ↔ ∪ (TC‘𝐴) ⊆ (TC‘𝐴))
22	20, 21	mpbi 230	. . . . . . . 8 ⊢ ∪ (TC‘𝐴) ⊆ (TC‘𝐴)
23	1	unisn 4890	. . . . . . . . 9 ⊢ ∪ {𝐴} = 𝐴
24		tcid 9692	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → 𝐴 ⊆ (TC‘𝐴))
25	1, 24	ax-mp 5	. . . . . . . . 9 ⊢ 𝐴 ⊆ (TC‘𝐴)
26	23, 25	eqsstri 3993	. . . . . . . 8 ⊢ ∪ {𝐴} ⊆ (TC‘𝐴)
27	22, 26	unssi 4154	. . . . . . 7 ⊢ (∪ (TC‘𝐴) ∪ ∪ {𝐴}) ⊆ (TC‘𝐴)
28	19, 27	eqsstri 3993	. . . . . 6 ⊢ ∪ ((TC‘𝐴) ∪ {𝐴}) ⊆ (TC‘𝐴)
29		ssun1 4141	. . . . . 6 ⊢ (TC‘𝐴) ⊆ ((TC‘𝐴) ∪ {𝐴})
30	28, 29	sstri 3956	. . . . 5 ⊢ ∪ ((TC‘𝐴) ∪ {𝐴}) ⊆ ((TC‘𝐴) ∪ {𝐴})
31		df-tr 5215	. . . . 5 ⊢ (Tr ((TC‘𝐴) ∪ {𝐴}) ↔ ∪ ((TC‘𝐴) ∪ {𝐴}) ⊆ ((TC‘𝐴) ∪ {𝐴}))
32	30, 31	mpbir 231	. . . 4 ⊢ Tr ((TC‘𝐴) ∪ {𝐴})
33		fvex 6871	. . . . . 6 ⊢ (TC‘𝐴) ∈ V
34		snex 5391	. . . . . 6 ⊢ {𝐴} ∈ V
35	33, 34	unex 7720	. . . . 5 ⊢ ((TC‘𝐴) ∪ {𝐴}) ∈ V
36		eleq2 2817	. . . . . 6 ⊢ (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ((TC‘𝐴) ∪ {𝐴})))
37		treq 5222	. . . . . 6 ⊢ (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → (Tr 𝑥 ↔ Tr ((TC‘𝐴) ∪ {𝐴})))
38	36, 37	anbi12d 632	. . . . 5 ⊢ (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → ((𝐴 ∈ 𝑥 ∧ Tr 𝑥) ↔ (𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}) ∧ Tr ((TC‘𝐴) ∪ {𝐴}))))
39	35, 38	elab 3646	. . . 4 ⊢ (((TC‘𝐴) ∪ {𝐴}) ∈ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)} ↔ (𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}) ∧ Tr ((TC‘𝐴) ∪ {𝐴})))
40	18, 32, 39	mpbir2an 711	. . 3 ⊢ ((TC‘𝐴) ∪ {𝐴}) ∈ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}
41		intss1 4927	. . 3 ⊢ (((TC‘𝐴) ∪ {𝐴}) ∈ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)} → ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)} ⊆ ((TC‘𝐴) ∪ {𝐴}))
42	40, 41	ax-mp 5	. 2 ⊢ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)} ⊆ ((TC‘𝐴) ∪ {𝐴})
43	15, 42	eqssi 3963	1 ⊢ ((TC‘𝐴) ∪ {𝐴}) = ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  Vcvv 3447   ∪ cun 3912   ⊆ wss 3914  {csn 4589  ∪ cuni 4871  ∩ cint 4910  Tr wtr 5214  ‘cfv 6511  TCctc 9689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711  ax-inf2 9594

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-tc 9690

This theorem is referenced by:  tcsni  9696