Theorem itunitc1 10460

Description: Each union iterate is a member of the transitive closure. (Contributed by Stefan O'Rear, 11-Feb-2015.)

Hypothesis

Ref	Expression
ituni.u	⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω))

Assertion

Ref	Expression
itunitc1	⊢ ((𝑈‘𝐴)‘𝐵) ⊆ (TC‘𝐴)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Allowed substitution hints:   𝑈(𝑥,𝑦)

Proof of Theorem itunitc1

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6906	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑈‘𝑎) = (𝑈‘𝐴))
2	1	fveq1d 6908	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑈‘𝑎)‘𝐵) = ((𝑈‘𝐴)‘𝐵))
3		fveq2 6906	. . . 4 ⊢ (𝑎 = 𝐴 → (TC‘𝑎) = (TC‘𝐴))
4	2, 3	sseq12d 4017	. . 3 ⊢ (𝑎 = 𝐴 → (((𝑈‘𝑎)‘𝐵) ⊆ (TC‘𝑎) ↔ ((𝑈‘𝐴)‘𝐵) ⊆ (TC‘𝐴)))
5		fveq2 6906	. . . . . 6 ⊢ (𝑏 = ∅ → ((𝑈‘𝑎)‘𝑏) = ((𝑈‘𝑎)‘∅))
6	5	sseq1d 4015	. . . . 5 ⊢ (𝑏 = ∅ → (((𝑈‘𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈‘𝑎)‘∅) ⊆ (TC‘𝑎)))
7		fveq2 6906	. . . . . 6 ⊢ (𝑏 = 𝑐 → ((𝑈‘𝑎)‘𝑏) = ((𝑈‘𝑎)‘𝑐))
8	7	sseq1d 4015	. . . . 5 ⊢ (𝑏 = 𝑐 → (((𝑈‘𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎)))
9		fveq2 6906	. . . . . 6 ⊢ (𝑏 = suc 𝑐 → ((𝑈‘𝑎)‘𝑏) = ((𝑈‘𝑎)‘suc 𝑐))
10	9	sseq1d 4015	. . . . 5 ⊢ (𝑏 = suc 𝑐 → (((𝑈‘𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈‘𝑎)‘suc 𝑐) ⊆ (TC‘𝑎)))
11		fveq2 6906	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝑈‘𝑎)‘𝑏) = ((𝑈‘𝑎)‘𝐵))
12	11	sseq1d 4015	. . . . 5 ⊢ (𝑏 = 𝐵 → (((𝑈‘𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈‘𝑎)‘𝐵) ⊆ (TC‘𝑎)))
13		ituni.u	. . . . . . . 8 ⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω))
14	13	ituni0 10458	. . . . . . 7 ⊢ (𝑎 ∈ V → ((𝑈‘𝑎)‘∅) = 𝑎)
15		tcid 9779	. . . . . . 7 ⊢ (𝑎 ∈ V → 𝑎 ⊆ (TC‘𝑎))
16	14, 15	eqsstrd 4018	. . . . . 6 ⊢ (𝑎 ∈ V → ((𝑈‘𝑎)‘∅) ⊆ (TC‘𝑎))
17	16	elv 3485	. . . . 5 ⊢ ((𝑈‘𝑎)‘∅) ⊆ (TC‘𝑎)
18	13	itunisuc 10459	. . . . . . 7 ⊢ ((𝑈‘𝑎)‘suc 𝑐) = ∪ ((𝑈‘𝑎)‘𝑐)
19		tctr 9780	. . . . . . . . . 10 ⊢ Tr (TC‘𝑎)
20		pwtr 5457	. . . . . . . . . 10 ⊢ (Tr (TC‘𝑎) ↔ Tr 𝒫 (TC‘𝑎))
21	19, 20	mpbi 230	. . . . . . . . 9 ⊢ Tr 𝒫 (TC‘𝑎)
22		trss 5270	. . . . . . . . 9 ⊢ (Tr 𝒫 (TC‘𝑎) → (((𝑈‘𝑎)‘𝑐) ∈ 𝒫 (TC‘𝑎) → ((𝑈‘𝑎)‘𝑐) ⊆ 𝒫 (TC‘𝑎)))
23	21, 22	ax-mp 5	. . . . . . . 8 ⊢ (((𝑈‘𝑎)‘𝑐) ∈ 𝒫 (TC‘𝑎) → ((𝑈‘𝑎)‘𝑐) ⊆ 𝒫 (TC‘𝑎))
24		fvex 6919	. . . . . . . . 9 ⊢ ((𝑈‘𝑎)‘𝑐) ∈ V
25	24	elpw 4604	. . . . . . . 8 ⊢ (((𝑈‘𝑎)‘𝑐) ∈ 𝒫 (TC‘𝑎) ↔ ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎))
26		sspwuni 5100	. . . . . . . 8 ⊢ (((𝑈‘𝑎)‘𝑐) ⊆ 𝒫 (TC‘𝑎) ↔ ∪ ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎))
27	23, 25, 26	3imtr3i 291	. . . . . . 7 ⊢ (((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎) → ∪ ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎))
28	18, 27	eqsstrid 4022	. . . . . 6 ⊢ (((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎) → ((𝑈‘𝑎)‘suc 𝑐) ⊆ (TC‘𝑎))
29	28	a1i 11	. . . . 5 ⊢ (𝑐 ∈ ω → (((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎) → ((𝑈‘𝑎)‘suc 𝑐) ⊆ (TC‘𝑎)))
30	6, 8, 10, 12, 17, 29	finds 7918	. . . 4 ⊢ (𝐵 ∈ ω → ((𝑈‘𝑎)‘𝐵) ⊆ (TC‘𝑎))
31		vex 3484	. . . . . . . 8 ⊢ 𝑎 ∈ V
32	13	itunifn 10457	. . . . . . . 8 ⊢ (𝑎 ∈ V → (𝑈‘𝑎) Fn ω)
33		fndm 6671	. . . . . . . 8 ⊢ ((𝑈‘𝑎) Fn ω → dom (𝑈‘𝑎) = ω)
34	31, 32, 33	mp2b 10	. . . . . . 7 ⊢ dom (𝑈‘𝑎) = ω
35	34	eleq2i 2833	. . . . . 6 ⊢ (𝐵 ∈ dom (𝑈‘𝑎) ↔ 𝐵 ∈ ω)
36		ndmfv 6941	. . . . . 6 ⊢ (¬ 𝐵 ∈ dom (𝑈‘𝑎) → ((𝑈‘𝑎)‘𝐵) = ∅)
37	35, 36	sylnbir 331	. . . . 5 ⊢ (¬ 𝐵 ∈ ω → ((𝑈‘𝑎)‘𝐵) = ∅)
38		0ss 4400	. . . . 5 ⊢ ∅ ⊆ (TC‘𝑎)
39	37, 38	eqsstrdi 4028	. . . 4 ⊢ (¬ 𝐵 ∈ ω → ((𝑈‘𝑎)‘𝐵) ⊆ (TC‘𝑎))
40	30, 39	pm2.61i 182	. . 3 ⊢ ((𝑈‘𝑎)‘𝐵) ⊆ (TC‘𝑎)
41	4, 40	vtoclg 3554	. 2 ⊢ (𝐴 ∈ V → ((𝑈‘𝐴)‘𝐵) ⊆ (TC‘𝐴))
42		fv2prc 6951	. . 3 ⊢ (¬ 𝐴 ∈ V → ((𝑈‘𝐴)‘𝐵) = ∅)
43		0ss 4400	. . 3 ⊢ ∅ ⊆ (TC‘𝐴)
44	42, 43	eqsstrdi 4028	. 2 ⊢ (¬ 𝐴 ∈ V → ((𝑈‘𝐴)‘𝐵) ⊆ (TC‘𝐴))
45	41, 44	pm2.61i 182	1 ⊢ ((𝑈‘𝐴)‘𝐵) ⊆ (TC‘𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2108  Vcvv 3480   ⊆ wss 3951  ∅c0 4333  𝒫 cpw 4600  ∪ cuni 4907   ↦ cmpt 5225  Tr wtr 5259  dom cdm 5685   ↾ cres 5687  suc csuc 6386   Fn wfn 6556  ‘cfv 6561  ωcom 7887  reccrdg 8449  TCctc 9776

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-inf2 9681

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-tc 9777

This theorem is referenced by:  itunitc  10461