Theorem tc2 8618

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 8614	. . . . 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 4761	. . . . . . 7 ⊢ (Tr 𝑥 → (𝐴 ∈ 𝑥 → 𝐴 ⊆ 𝑥))
5	4	imdistanri 727	. . . . . 6 ⊢ ((𝐴 ∈ 𝑥 ∧ Tr 𝑥) → (𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
6	5	ss2abi 3674	. . . . 5 ⊢ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}
7		intss 4498	. . . . 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 3635	. . 3 ⊢ (TC‘𝐴) ⊆ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}
10	1	elintab 4487	. . . . 5 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)} ↔ ∀𝑥((𝐴 ∈ 𝑥 ∧ Tr 𝑥) → 𝐴 ∈ 𝑥))
11		simpl 473	. . . . 5 ⊢ ((𝐴 ∈ 𝑥 ∧ Tr 𝑥) → 𝐴 ∈ 𝑥)
12	10, 11	mpgbir 1726	. . . 4 ⊢ 𝐴 ∈ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}
13	1	snss 4316	. . . 4 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)} ↔ {𝐴} ⊆ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)})
14	12, 13	mpbi 220	. . 3 ⊢ {𝐴} ⊆ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}
15	9, 14	unssi 3788	. 2 ⊢ ((TC‘𝐴) ∪ {𝐴}) ⊆ ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}
16	1	snid 4208	. . . . 5 ⊢ 𝐴 ∈ {𝐴}
17		elun2 3781	. . . . 5 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}))
18	16, 17	ax-mp 5	. . . 4 ⊢ 𝐴 ∈ ((TC‘𝐴) ∪ {𝐴})
19		uniun 4456	. . . . . . 7 ⊢ ∪ ((TC‘𝐴) ∪ {𝐴}) = (∪ (TC‘𝐴) ∪ ∪ {𝐴})
20		tctr 8616	. . . . . . . . 9 ⊢ Tr (TC‘𝐴)
21		df-tr 4753	. . . . . . . . 9 ⊢ (Tr (TC‘𝐴) ↔ ∪ (TC‘𝐴) ⊆ (TC‘𝐴))
22	20, 21	mpbi 220	. . . . . . . 8 ⊢ ∪ (TC‘𝐴) ⊆ (TC‘𝐴)
23	1	unisn 4451	. . . . . . . . 9 ⊢ ∪ {𝐴} = 𝐴
24		tcid 8615	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → 𝐴 ⊆ (TC‘𝐴))
25	1, 24	ax-mp 5	. . . . . . . . 9 ⊢ 𝐴 ⊆ (TC‘𝐴)
26	23, 25	eqsstri 3635	. . . . . . . 8 ⊢ ∪ {𝐴} ⊆ (TC‘𝐴)
27	22, 26	unssi 3788	. . . . . . 7 ⊢ (∪ (TC‘𝐴) ∪ ∪ {𝐴}) ⊆ (TC‘𝐴)
28	19, 27	eqsstri 3635	. . . . . 6 ⊢ ∪ ((TC‘𝐴) ∪ {𝐴}) ⊆ (TC‘𝐴)
29		ssun1 3776	. . . . . 6 ⊢ (TC‘𝐴) ⊆ ((TC‘𝐴) ∪ {𝐴})
30	28, 29	sstri 3612	. . . . 5 ⊢ ∪ ((TC‘𝐴) ∪ {𝐴}) ⊆ ((TC‘𝐴) ∪ {𝐴})
31		df-tr 4753	. . . . 5 ⊢ (Tr ((TC‘𝐴) ∪ {𝐴}) ↔ ∪ ((TC‘𝐴) ∪ {𝐴}) ⊆ ((TC‘𝐴) ∪ {𝐴}))
32	30, 31	mpbir 221	. . . 4 ⊢ Tr ((TC‘𝐴) ∪ {𝐴})
33		fvex 6201	. . . . . 6 ⊢ (TC‘𝐴) ∈ V
34		snex 4908	. . . . . 6 ⊢ {𝐴} ∈ V
35	33, 34	unex 6956	. . . . 5 ⊢ ((TC‘𝐴) ∪ {𝐴}) ∈ V
36		eleq2 2690	. . . . . 6 ⊢ (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ((TC‘𝐴) ∪ {𝐴})))
37		treq 4758	. . . . . 6 ⊢ (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → (Tr 𝑥 ↔ Tr ((TC‘𝐴) ∪ {𝐴})))
38	36, 37	anbi12d 747	. . . . 5 ⊢ (𝑥 = ((TC‘𝐴) ∪ {𝐴}) → ((𝐴 ∈ 𝑥 ∧ Tr 𝑥) ↔ (𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}) ∧ Tr ((TC‘𝐴) ∪ {𝐴}))))
39	35, 38	elab 3350	. . . 4 ⊢ (((TC‘𝐴) ∪ {𝐴}) ∈ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)} ↔ (𝐴 ∈ ((TC‘𝐴) ∪ {𝐴}) ∧ Tr ((TC‘𝐴) ∪ {𝐴})))
40	18, 32, 39	mpbir2an 955	. . 3 ⊢ ((TC‘𝐴) ∪ {𝐴}) ∈ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}
41		intss1 4492	. . 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 3619	1 ⊢ ((TC‘𝐴) ∪ {𝐴}) = ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  Vcvv 3200   ∪ cun 3572   ⊆ wss 3574  {csn 4177  ∪ cuni 4436  ∩ cint 4475  Tr wtr 4752  ‘cfv 5888  TCctc 8612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-tc 8613

This theorem is referenced by:  tcsni  8619