Theorem tc2 9661

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 9657	. . . . 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 5202	. . . . . . 7 ⊢ (Tr 𝑥 → (𝐴 ∈ 𝑥 → 𝐴 ⊆ 𝑥))
5	4	imdistanri 569	. . . . . 6 ⊢ ((𝐴 ∈ 𝑥 ∧ Tr 𝑥) → (𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
6	5	ss2abi 4006	. . . . 5 ⊢ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}
7		intss 4911	. . . . 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 4901	. . . . 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 1801	. . . 4 ⊢ 𝐴 ∈ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}
13	1	snss 4728	. . . 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 4131	. 2 ⊢ ((TC‘𝐴) ∪ {𝐴}) ⊆ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}
16	1	snid 4606	. . . . 5 ⊢ 𝐴 ∈ {𝐴}
17		elun2 4123	. . . . 5 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}))
18	16, 17	ax-mp 5	. . . 4 ⊢ 𝐴 ∈ ((TC‘𝐴) ∪ {𝐴})
19		uniun 4873	. . . . . . 7 ⊢ ∪ ((TC‘𝐴) ∪ {𝐴}) = (∪ (TC‘𝐴) ∪ ∪ {𝐴})
20		tctr 9659	. . . . . . . . 9 ⊢ Tr (TC‘𝐴)
21		df-tr 5193	. . . . . . . . 9 ⊢ (Tr (TC‘𝐴) ↔ ∪ (TC‘𝐴) ⊆ (TC‘𝐴))
22	20, 21	mpbi 230	. . . . . . . 8 ⊢ ∪ (TC‘𝐴) ⊆ (TC‘𝐴)
23	1	unisn 4869	. . . . . . . . 9 ⊢ ∪ {𝐴} = 𝐴
24		tcid 9658	. . . . . . . . . 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 4131	. . . . . . 7 ⊢ (∪ (TC‘𝐴) ∪ ∪ {𝐴}) ⊆ (TC‘𝐴)
28	19, 27	eqsstri 3968	. . . . . 6 ⊢ ∪ ((TC‘𝐴) ∪ {𝐴}) ⊆ (TC‘𝐴)
29		ssun1 4118	. . . . . 6 ⊢ (TC‘𝐴) ⊆ ((TC‘𝐴) ∪ {𝐴})
30	28, 29	sstri 3931	. . . . 5 ⊢ ∪ ((TC‘𝐴) ∪ {𝐴}) ⊆ ((TC‘𝐴) ∪ {𝐴})
31		df-tr 5193	. . . . 5 ⊢ (Tr ((TC‘𝐴) ∪ {𝐴}) ↔ ∪ ((TC‘𝐴) ∪ {𝐴}) ⊆ ((TC‘𝐴) ∪ {𝐴}))
32	30, 31	mpbir 231	. . . 4 ⊢ Tr ((TC‘𝐴) ∪ {𝐴})
33		fvex 6853	. . . . . 6 ⊢ (TC‘𝐴) ∈ V
34		snex 5381	. . . . . 6 ⊢ {𝐴} ∈ V
35	33, 34	unex 7698	. . . . 5 ⊢ ((TC‘𝐴) ∪ {𝐴}) ∈ V
36		eleq2 2825	. . . . . 6 ⊢ (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ((TC‘𝐴) ∪ {𝐴})))
37		treq 5199	. . . . . 6 ⊢ (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → (Tr 𝑥 ↔ Tr ((TC‘𝐴) ∪ {𝐴})))
38	36, 37	anbi12d 633	. . . . 5 ⊢ (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → ((𝐴 ∈ 𝑥 ∧ Tr 𝑥) ↔ (𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}) ∧ Tr ((TC‘𝐴) ∪ {𝐴}))))
39	35, 38	elab 3622	. . . 4 ⊢ (((TC‘𝐴) ∪ {𝐴}) ∈ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)} ↔ (𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}) ∧ Tr ((TC‘𝐴) ∪ {𝐴})))
40	18, 32, 39	mpbir2an 712	. . 3 ⊢ ((TC‘𝐴) ∪ {𝐴}) ∈ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}
41		intss1 4905	. . 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 395   = wceq 1542   ∈ wcel 2114  {cab 2714  Vcvv 3429   ∪ cun 3887   ⊆ wss 3889  {csn 4567  ∪ cuni 4850  ∩ cint 4889  Tr wtr 5192  ‘cfv 6498  TCctc 9655

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689  ax-inf2 9562

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-tc 9656

This theorem is referenced by:  tcsni  9662