Theorem itunitc1 10489

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 6920	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑈‘𝑎) = (𝑈‘𝐴))
2	1	fveq1d 6922	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑈‘𝑎)‘𝐵) = ((𝑈‘𝐴)‘𝐵))
3		fveq2 6920	. . . 4 ⊢ (𝑎 = 𝐴 → (TC‘𝑎) = (TC‘𝐴))
4	2, 3	sseq12d 4042	. . 3 ⊢ (𝑎 = 𝐴 → (((𝑈‘𝑎)‘𝐵) ⊆ (TC‘𝑎) ↔ ((𝑈‘𝐴)‘𝐵) ⊆ (TC‘𝐴)))
5		fveq2 6920	. . . . . 6 ⊢ (𝑏 = ∅ → ((𝑈‘𝑎)‘𝑏) = ((𝑈‘𝑎)‘∅))
6	5	sseq1d 4040	. . . . 5 ⊢ (𝑏 = ∅ → (((𝑈‘𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈‘𝑎)‘∅) ⊆ (TC‘𝑎)))
7		fveq2 6920	. . . . . 6 ⊢ (𝑏 = 𝑐 → ((𝑈‘𝑎)‘𝑏) = ((𝑈‘𝑎)‘𝑐))
8	7	sseq1d 4040	. . . . 5 ⊢ (𝑏 = 𝑐 → (((𝑈‘𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎)))
9		fveq2 6920	. . . . . 6 ⊢ (𝑏 = suc 𝑐 → ((𝑈‘𝑎)‘𝑏) = ((𝑈‘𝑎)‘suc 𝑐))
10	9	sseq1d 4040	. . . . 5 ⊢ (𝑏 = suc 𝑐 → (((𝑈‘𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈‘𝑎)‘suc 𝑐) ⊆ (TC‘𝑎)))
11		fveq2 6920	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝑈‘𝑎)‘𝑏) = ((𝑈‘𝑎)‘𝐵))
12	11	sseq1d 4040	. . . . 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 10487	. . . . . . 7 ⊢ (𝑎 ∈ V → ((𝑈‘𝑎)‘∅) = 𝑎)
15		tcid 9808	. . . . . . 7 ⊢ (𝑎 ∈ V → 𝑎 ⊆ (TC‘𝑎))
16	14, 15	eqsstrd 4047	. . . . . 6 ⊢ (𝑎 ∈ V → ((𝑈‘𝑎)‘∅) ⊆ (TC‘𝑎))
17	16	elv 3493	. . . . 5 ⊢ ((𝑈‘𝑎)‘∅) ⊆ (TC‘𝑎)
18	13	itunisuc 10488	. . . . . . 7 ⊢ ((𝑈‘𝑎)‘suc 𝑐) = ∪ ((𝑈‘𝑎)‘𝑐)
19		tctr 9809	. . . . . . . . . 10 ⊢ Tr (TC‘𝑎)
20		pwtr 5472	. . . . . . . . . 10 ⊢ (Tr (TC‘𝑎) ↔ Tr 𝒫 (TC‘𝑎))
21	19, 20	mpbi 230	. . . . . . . . 9 ⊢ Tr 𝒫 (TC‘𝑎)
22		trss 5294	. . . . . . . . 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 6933	. . . . . . . . 9 ⊢ ((𝑈‘𝑎)‘𝑐) ∈ V
25	24	elpw 4626	. . . . . . . 8 ⊢ (((𝑈‘𝑎)‘𝑐) ∈ 𝒫 (TC‘𝑎) ↔ ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎))
26		sspwuni 5123	. . . . . . . 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 4057	. . . . . 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 7936	. . . 4 ⊢ (𝐵 ∈ ω → ((𝑈‘𝑎)‘𝐵) ⊆ (TC‘𝑎))
31		vex 3492	. . . . . . . 8 ⊢ 𝑎 ∈ V
32	13	itunifn 10486	. . . . . . . 8 ⊢ (𝑎 ∈ V → (𝑈‘𝑎) Fn ω)
33		fndm 6682	. . . . . . . 8 ⊢ ((𝑈‘𝑎) Fn ω → dom (𝑈‘𝑎) = ω)
34	31, 32, 33	mp2b 10	. . . . . . 7 ⊢ dom (𝑈‘𝑎) = ω
35	34	eleq2i 2836	. . . . . 6 ⊢ (𝐵 ∈ dom (𝑈‘𝑎) ↔ 𝐵 ∈ ω)
36		ndmfv 6955	. . . . . 6 ⊢ (¬ 𝐵 ∈ dom (𝑈‘𝑎) → ((𝑈‘𝑎)‘𝐵) = ∅)
37	35, 36	sylnbir 331	. . . . 5 ⊢ (¬ 𝐵 ∈ ω → ((𝑈‘𝑎)‘𝐵) = ∅)
38		0ss 4423	. . . . 5 ⊢ ∅ ⊆ (TC‘𝑎)
39	37, 38	eqsstrdi 4063	. . . 4 ⊢ (¬ 𝐵 ∈ ω → ((𝑈‘𝑎)‘𝐵) ⊆ (TC‘𝑎))
40	30, 39	pm2.61i 182	. . 3 ⊢ ((𝑈‘𝑎)‘𝐵) ⊆ (TC‘𝑎)
41	4, 40	vtoclg 3566	. 2 ⊢ (𝐴 ∈ V → ((𝑈‘𝐴)‘𝐵) ⊆ (TC‘𝐴))
42		fv2prc 6965	. . 3 ⊢ (¬ 𝐴 ∈ V → ((𝑈‘𝐴)‘𝐵) = ∅)
43		0ss 4423	. . 3 ⊢ ∅ ⊆ (TC‘𝐴)
44	42, 43	eqsstrdi 4063	. 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 1537   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  ∪ cuni 4931   ↦ cmpt 5249  Tr wtr 5283  dom cdm 5700   ↾ cres 5702  suc csuc 6397   Fn wfn 6568  ‘cfv 6573  ωcom 7903  reccrdg 8465  TCctc 9805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770  ax-inf2 9710

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-tc 9806

This theorem is referenced by:  itunitc  10490