Theorem tc2 9659

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 9655	. . . . 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 5196	. . . . . . 7 ⊢ (Tr 𝑥 → (𝐴 ∈ 𝑥 → 𝐴 ⊆ 𝑥))
5	4	imdistanri 574	. . . . . 6 ⊢ ((𝐴 ∈ 𝑥 ∧ Tr 𝑥) → (𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
6	5	ss2abi 4004	. . . . 5 ⊢ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}
7		intss 4906	. . . . 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 3968	. . 3 ⊢ (TC‘𝐴) ⊆ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}
10	1	elintab 4896	. . . . 5 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)} ↔ ∀𝑥((𝐴 ∈ 𝑥 ∧ Tr 𝑥) → 𝐴 ∈ 𝑥))
11		simpl 483	. . . . 5 ⊢ ((𝐴 ∈ 𝑥 ∧ Tr 𝑥) → 𝐴 ∈ 𝑥)
12	10, 11	mpgbir 1806	. . . 4 ⊢ 𝐴 ∈ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}
13	1	snss 4723	. . . 4 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)} ↔ {𝐴} ⊆ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)})
14	12, 13	mpbi 231	. . 3 ⊢ {𝐴} ⊆ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}
15	9, 14	unssi 4127	. 2 ⊢ ((TC‘𝐴) ∪ {𝐴}) ⊆ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}
16	1	snid 4601	. . . . 5 ⊢ 𝐴 ∈ {𝐴}
17		elun2 4119	. . . . 5 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}))
18	16, 17	ax-mp 5	. . . 4 ⊢ 𝐴 ∈ ((TC‘𝐴) ∪ {𝐴})
19		uniun 4868	. . . . . . 7 ⊢ ∪ ((TC‘𝐴) ∪ {𝐴}) = (∪ (TC‘𝐴) ∪ ∪ {𝐴})
20		tctr 9657	. . . . . . . . 9 ⊢ Tr (TC‘𝐴)
21		df-tr 5187	. . . . . . . . 9 ⊢ (Tr (TC‘𝐴) ↔ ∪ (TC‘𝐴) ⊆ (TC‘𝐴))
22	20, 21	mpbi 231	. . . . . . . 8 ⊢ ∪ (TC‘𝐴) ⊆ (TC‘𝐴)
23	1	unisn 4864	. . . . . . . . 9 ⊢ ∪ {𝐴} = 𝐴
24		tcid 9656	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → 𝐴 ⊆ (TC‘𝐴))
25	1, 24	ax-mp 5	. . . . . . . . 9 ⊢ 𝐴 ⊆ (TC‘𝐴)
26	23, 25	eqsstri 3968	. . . . . . . 8 ⊢ ∪ {𝐴} ⊆ (TC‘𝐴)
27	22, 26	unssi 4127	. . . . . . 7 ⊢ (∪ (TC‘𝐴) ∪ ∪ {𝐴}) ⊆ (TC‘𝐴)
28	19, 27	eqsstri 3968	. . . . . 6 ⊢ ∪ ((TC‘𝐴) ∪ {𝐴}) ⊆ (TC‘𝐴)
29		ssun1 4114	. . . . . 6 ⊢ (TC‘𝐴) ⊆ ((TC‘𝐴) ∪ {𝐴})
30	28, 29	sstri 3931	. . . . 5 ⊢ ∪ ((TC‘𝐴) ∪ {𝐴}) ⊆ ((TC‘𝐴) ∪ {𝐴})
31		df-tr 5187	. . . . 5 ⊢ (Tr ((TC‘𝐴) ∪ {𝐴}) ↔ ∪ ((TC‘𝐴) ∪ {𝐴}) ⊆ ((TC‘𝐴) ∪ {𝐴}))
32	30, 31	mpbir 232	. . . 4 ⊢ Tr ((TC‘𝐴) ∪ {𝐴})
33		fvex 6847	. . . . . 6 ⊢ (TC‘𝐴) ∈ V
34		snex 5375	. . . . . 6 ⊢ {𝐴} ∈ V
35	33, 34	unex 7694	. . . . 5 ⊢ ((TC‘𝐴) ∪ {𝐴}) ∈ V
36		eleq2 2829	. . . . . 6 ⊢ (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ((TC‘𝐴) ∪ {𝐴})))
37		treq 5193	. . . . . 6 ⊢ (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → (Tr 𝑥 ↔ Tr ((TC‘𝐴) ∪ {𝐴})))
38	36, 37	anbi12d 638	. . . . 5 ⊢ (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → ((𝐴 ∈ 𝑥 ∧ Tr 𝑥) ↔ (𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}) ∧ Tr ((TC‘𝐴) ∪ {𝐴}))))
39	35, 38	elab 3624	. . . 4 ⊢ (((TC‘𝐴) ∪ {𝐴}) ∈ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)} ↔ (𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}) ∧ Tr ((TC‘𝐴) ∪ {𝐴})))
40	18, 32, 39	mpbir2an 717	. . 3 ⊢ ((TC‘𝐴) ∪ {𝐴}) ∈ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}
41		intss1 4900	. . 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 3938	1 ⊢ ((TC‘𝐴) ∪ {𝐴}) = ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {cab 2718  Vcvv 3432   ∪ cun 3888   ⊆ wss 3890  {csn 4562  ∪ cuni 4845  ∩ cint 4884  Tr wtr 5186  ‘cfv 6492  TCctc 9653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685  ax-inf2 9560

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-iin 4931  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  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 7366  df-om 7814  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-tc 9654

This theorem is referenced by:  tcsni  9660