Theorem itunitc1 10342

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 6841	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑈‘𝑎) = (𝑈‘𝐴))
2	1	fveq1d 6843	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑈‘𝑎)‘𝐵) = ((𝑈‘𝐴)‘𝐵))
3		fveq2 6841	. . . 4 ⊢ (𝑎 = 𝐴 → (TC‘𝑎) = (TC‘𝐴))
4	2, 3	sseq12d 3956	. . 3 ⊢ (𝑎 = 𝐴 → (((𝑈‘𝑎)‘𝐵) ⊆ (TC‘𝑎) ↔ ((𝑈‘𝐴)‘𝐵) ⊆ (TC‘𝐴)))
5		fveq2 6841	. . . . . 6 ⊢ (𝑏 = ∅ → ((𝑈‘𝑎)‘𝑏) = ((𝑈‘𝑎)‘∅))
6	5	sseq1d 3954	. . . . 5 ⊢ (𝑏 = ∅ → (((𝑈‘𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈‘𝑎)‘∅) ⊆ (TC‘𝑎)))
7		fveq2 6841	. . . . . 6 ⊢ (𝑏 = 𝑐 → ((𝑈‘𝑎)‘𝑏) = ((𝑈‘𝑎)‘𝑐))
8	7	sseq1d 3954	. . . . 5 ⊢ (𝑏 = 𝑐 → (((𝑈‘𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎)))
9		fveq2 6841	. . . . . 6 ⊢ (𝑏 = suc 𝑐 → ((𝑈‘𝑎)‘𝑏) = ((𝑈‘𝑎)‘suc 𝑐))
10	9	sseq1d 3954	. . . . 5 ⊢ (𝑏 = suc 𝑐 → (((𝑈‘𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈‘𝑎)‘suc 𝑐) ⊆ (TC‘𝑎)))
11		fveq2 6841	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝑈‘𝑎)‘𝑏) = ((𝑈‘𝑎)‘𝐵))
12	11	sseq1d 3954	. . . . 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 10340	. . . . . . 7 ⊢ (𝑎 ∈ V → ((𝑈‘𝑎)‘∅) = 𝑎)
15		tcid 9658	. . . . . . 7 ⊢ (𝑎 ∈ V → 𝑎 ⊆ (TC‘𝑎))
16	14, 15	eqsstrd 3957	. . . . . 6 ⊢ (𝑎 ∈ V → ((𝑈‘𝑎)‘∅) ⊆ (TC‘𝑎))
17	16	elv 3435	. . . . 5 ⊢ ((𝑈‘𝑎)‘∅) ⊆ (TC‘𝑎)
18	13	itunisuc 10341	. . . . . . 7 ⊢ ((𝑈‘𝑎)‘suc 𝑐) = ∪ ((𝑈‘𝑎)‘𝑐)
19		tctr 9659	. . . . . . . . . 10 ⊢ Tr (TC‘𝑎)
20		pwtr 5405	. . . . . . . . . 10 ⊢ (Tr (TC‘𝑎) ↔ Tr 𝒫 (TC‘𝑎))
21	19, 20	mpbi 230	. . . . . . . . 9 ⊢ Tr 𝒫 (TC‘𝑎)
22		trss 5203	. . . . . . . . 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 6854	. . . . . . . . 9 ⊢ ((𝑈‘𝑎)‘𝑐) ∈ V
25	24	elpw 4546	. . . . . . . 8 ⊢ (((𝑈‘𝑎)‘𝑐) ∈ 𝒫 (TC‘𝑎) ↔ ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎))
26		sspwuni 5043	. . . . . . . 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 3961	. . . . . 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 7847	. . . 4 ⊢ (𝐵 ∈ ω → ((𝑈‘𝑎)‘𝐵) ⊆ (TC‘𝑎))
31		vex 3434	. . . . . . . 8 ⊢ 𝑎 ∈ V
32	13	itunifn 10339	. . . . . . . 8 ⊢ (𝑎 ∈ V → (𝑈‘𝑎) Fn ω)
33		fndm 6602	. . . . . . . 8 ⊢ ((𝑈‘𝑎) Fn ω → dom (𝑈‘𝑎) = ω)
34	31, 32, 33	mp2b 10	. . . . . . 7 ⊢ dom (𝑈‘𝑎) = ω
35	34	eleq2i 2829	. . . . . 6 ⊢ (𝐵 ∈ dom (𝑈‘𝑎) ↔ 𝐵 ∈ ω)
36		ndmfv 6873	. . . . . 6 ⊢ (¬ 𝐵 ∈ dom (𝑈‘𝑎) → ((𝑈‘𝑎)‘𝐵) = ∅)
37	35, 36	sylnbir 331	. . . . 5 ⊢ (¬ 𝐵 ∈ ω → ((𝑈‘𝑎)‘𝐵) = ∅)
38		0ss 4341	. . . . 5 ⊢ ∅ ⊆ (TC‘𝑎)
39	37, 38	eqsstrdi 3967	. . . 4 ⊢ (¬ 𝐵 ∈ ω → ((𝑈‘𝑎)‘𝐵) ⊆ (TC‘𝑎))
40	30, 39	pm2.61i 182	. . 3 ⊢ ((𝑈‘𝑎)‘𝐵) ⊆ (TC‘𝑎)
41	4, 40	vtoclg 3500	. 2 ⊢ (𝐴 ∈ V → ((𝑈‘𝐴)‘𝐵) ⊆ (TC‘𝐴))
42		fv2prc 6883	. . 3 ⊢ (¬ 𝐴 ∈ V → ((𝑈‘𝐴)‘𝐵) = ∅)
43		0ss 4341	. . 3 ⊢ ∅ ⊆ (TC‘𝐴)
44	42, 43	eqsstrdi 3967	. 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 1542   ∈ wcel 2114  Vcvv 3430   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542  ∪ cuni 4851   ↦ cmpt 5167  Tr wtr 5193  dom cdm 5631   ↾ cres 5633  suc csuc 6326   Fn wfn 6494  ‘cfv 6499  ωcom 7817  reccrdg 8348  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 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pr 5376  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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  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 6266  df-ord 6327  df-on 6328  df-lim 6329  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  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:  itunitc  10343