Theorem itunitc1 9125

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 6103	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑈‘𝑎) = (𝑈‘𝐴))
2	1	fveq1d 6105	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑈‘𝑎)‘𝐵) = ((𝑈‘𝐴)‘𝐵))
3		fveq2 6103	. . . 4 ⊢ (𝑎 = 𝐴 → (TC‘𝑎) = (TC‘𝐴))
4	2, 3	sseq12d 3597	. . 3 ⊢ (𝑎 = 𝐴 → (((𝑈‘𝑎)‘𝐵) ⊆ (TC‘𝑎) ↔ ((𝑈‘𝐴)‘𝐵) ⊆ (TC‘𝐴)))
5		fveq2 6103	. . . . . 6 ⊢ (𝑏 = ∅ → ((𝑈‘𝑎)‘𝑏) = ((𝑈‘𝑎)‘∅))
6	5	sseq1d 3595	. . . . 5 ⊢ (𝑏 = ∅ → (((𝑈‘𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈‘𝑎)‘∅) ⊆ (TC‘𝑎)))
7		fveq2 6103	. . . . . 6 ⊢ (𝑏 = 𝑐 → ((𝑈‘𝑎)‘𝑏) = ((𝑈‘𝑎)‘𝑐))
8	7	sseq1d 3595	. . . . 5 ⊢ (𝑏 = 𝑐 → (((𝑈‘𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎)))
9		fveq2 6103	. . . . . 6 ⊢ (𝑏 = suc 𝑐 → ((𝑈‘𝑎)‘𝑏) = ((𝑈‘𝑎)‘suc 𝑐))
10	9	sseq1d 3595	. . . . 5 ⊢ (𝑏 = suc 𝑐 → (((𝑈‘𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈‘𝑎)‘suc 𝑐) ⊆ (TC‘𝑎)))
11		fveq2 6103	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝑈‘𝑎)‘𝑏) = ((𝑈‘𝑎)‘𝐵))
12	11	sseq1d 3595	. . . . 5 ⊢ (𝑏 = 𝐵 → (((𝑈‘𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈‘𝑎)‘𝐵) ⊆ (TC‘𝑎)))
13		vex 3176	. . . . . 6 ⊢ 𝑎 ∈ V
14		ituni.u	. . . . . . . 8 ⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω))
15	14	ituni0 9123	. . . . . . 7 ⊢ (𝑎 ∈ V → ((𝑈‘𝑎)‘∅) = 𝑎)
16		tcid 8498	. . . . . . 7 ⊢ (𝑎 ∈ V → 𝑎 ⊆ (TC‘𝑎))
17	15, 16	eqsstrd 3602	. . . . . 6 ⊢ (𝑎 ∈ V → ((𝑈‘𝑎)‘∅) ⊆ (TC‘𝑎))
18	13, 17	ax-mp 5	. . . . 5 ⊢ ((𝑈‘𝑎)‘∅) ⊆ (TC‘𝑎)
19	14	itunisuc 9124	. . . . . . 7 ⊢ ((𝑈‘𝑎)‘suc 𝑐) = ∪ ((𝑈‘𝑎)‘𝑐)
20		tctr 8499	. . . . . . . . . 10 ⊢ Tr (TC‘𝑎)
21		pwtr 4848	. . . . . . . . . 10 ⊢ (Tr (TC‘𝑎) ↔ Tr 𝒫 (TC‘𝑎))
22	20, 21	mpbi 219	. . . . . . . . 9 ⊢ Tr 𝒫 (TC‘𝑎)
23		trss 4689	. . . . . . . . 9 ⊢ (Tr 𝒫 (TC‘𝑎) → (((𝑈‘𝑎)‘𝑐) ∈ 𝒫 (TC‘𝑎) → ((𝑈‘𝑎)‘𝑐) ⊆ 𝒫 (TC‘𝑎)))
24	22, 23	ax-mp 5	. . . . . . . 8 ⊢ (((𝑈‘𝑎)‘𝑐) ∈ 𝒫 (TC‘𝑎) → ((𝑈‘𝑎)‘𝑐) ⊆ 𝒫 (TC‘𝑎))
25		fvex 6113	. . . . . . . . 9 ⊢ ((𝑈‘𝑎)‘𝑐) ∈ V
26	25	elpw 4114	. . . . . . . 8 ⊢ (((𝑈‘𝑎)‘𝑐) ∈ 𝒫 (TC‘𝑎) ↔ ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎))
27		sspwuni 4547	. . . . . . . 8 ⊢ (((𝑈‘𝑎)‘𝑐) ⊆ 𝒫 (TC‘𝑎) ↔ ∪ ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎))
28	24, 26, 27	3imtr3i 279	. . . . . . 7 ⊢ (((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎) → ∪ ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎))
29	19, 28	syl5eqss 3612	. . . . . 6 ⊢ (((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎) → ((𝑈‘𝑎)‘suc 𝑐) ⊆ (TC‘𝑎))
30	29	a1i 11	. . . . 5 ⊢ (𝑐 ∈ ω → (((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎) → ((𝑈‘𝑎)‘suc 𝑐) ⊆ (TC‘𝑎)))
31	6, 8, 10, 12, 18, 30	finds 6984	. . . 4 ⊢ (𝐵 ∈ ω → ((𝑈‘𝑎)‘𝐵) ⊆ (TC‘𝑎))
32	14	itunifn 9122	. . . . . . . 8 ⊢ (𝑎 ∈ V → (𝑈‘𝑎) Fn ω)
33		fndm 5904	. . . . . . . 8 ⊢ ((𝑈‘𝑎) Fn ω → dom (𝑈‘𝑎) = ω)
34	13, 32, 33	mp2b 10	. . . . . . 7 ⊢ dom (𝑈‘𝑎) = ω
35	34	eleq2i 2680	. . . . . 6 ⊢ (𝐵 ∈ dom (𝑈‘𝑎) ↔ 𝐵 ∈ ω)
36		ndmfv 6128	. . . . . 6 ⊢ (¬ 𝐵 ∈ dom (𝑈‘𝑎) → ((𝑈‘𝑎)‘𝐵) = ∅)
37	35, 36	sylnbir 320	. . . . 5 ⊢ (¬ 𝐵 ∈ ω → ((𝑈‘𝑎)‘𝐵) = ∅)
38		0ss 3924	. . . . 5 ⊢ ∅ ⊆ (TC‘𝑎)
39	37, 38	syl6eqss 3618	. . . 4 ⊢ (¬ 𝐵 ∈ ω → ((𝑈‘𝑎)‘𝐵) ⊆ (TC‘𝑎))
40	31, 39	pm2.61i 175	. . 3 ⊢ ((𝑈‘𝑎)‘𝐵) ⊆ (TC‘𝑎)
41	4, 40	vtoclg 3239	. 2 ⊢ (𝐴 ∈ V → ((𝑈‘𝐴)‘𝐵) ⊆ (TC‘𝐴))
42		fvprc 6097	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑈‘𝐴) = ∅)
43	42	fveq1d 6105	. . . 4 ⊢ (¬ 𝐴 ∈ V → ((𝑈‘𝐴)‘𝐵) = (∅‘𝐵))
44		0fv 6137	. . . 4 ⊢ (∅‘𝐵) = ∅
45	43, 44	syl6eq 2660	. . 3 ⊢ (¬ 𝐴 ∈ V → ((𝑈‘𝐴)‘𝐵) = ∅)
46		0ss 3924	. . 3 ⊢ ∅ ⊆ (TC‘𝐴)
47	45, 46	syl6eqss 3618	. 2 ⊢ (¬ 𝐴 ∈ V → ((𝑈‘𝐴)‘𝐵) ⊆ (TC‘𝐴))
48	41, 47	pm2.61i 175	1 ⊢ ((𝑈‘𝐴)‘𝐵) ⊆ (TC‘𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372   ↦ cmpt 4643  Tr wtr 4680  dom cdm 5038   ↾ cres 5040  suc csuc 5642   Fn wfn 5799  ‘cfv 5804  ωcom 6957  reccrdg 7392  TCctc 8495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-tc 8496

This theorem is referenced by:  itunitc  9126