Theorem itunitc1 9695

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 6545	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑈‘𝑎) = (𝑈‘𝐴))
2	1	fveq1d 6547	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑈‘𝑎)‘𝐵) = ((𝑈‘𝐴)‘𝐵))
3		fveq2 6545	. . . 4 ⊢ (𝑎 = 𝐴 → (TC‘𝑎) = (TC‘𝐴))
4	2, 3	sseq12d 3927	. . 3 ⊢ (𝑎 = 𝐴 → (((𝑈‘𝑎)‘𝐵) ⊆ (TC‘𝑎) ↔ ((𝑈‘𝐴)‘𝐵) ⊆ (TC‘𝐴)))
5		fveq2 6545	. . . . . 6 ⊢ (𝑏 = ∅ → ((𝑈‘𝑎)‘𝑏) = ((𝑈‘𝑎)‘∅))
6	5	sseq1d 3925	. . . . 5 ⊢ (𝑏 = ∅ → (((𝑈‘𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈‘𝑎)‘∅) ⊆ (TC‘𝑎)))
7		fveq2 6545	. . . . . 6 ⊢ (𝑏 = 𝑐 → ((𝑈‘𝑎)‘𝑏) = ((𝑈‘𝑎)‘𝑐))
8	7	sseq1d 3925	. . . . 5 ⊢ (𝑏 = 𝑐 → (((𝑈‘𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎)))
9		fveq2 6545	. . . . . 6 ⊢ (𝑏 = suc 𝑐 → ((𝑈‘𝑎)‘𝑏) = ((𝑈‘𝑎)‘suc 𝑐))
10	9	sseq1d 3925	. . . . 5 ⊢ (𝑏 = suc 𝑐 → (((𝑈‘𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈‘𝑎)‘suc 𝑐) ⊆ (TC‘𝑎)))
11		fveq2 6545	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝑈‘𝑎)‘𝑏) = ((𝑈‘𝑎)‘𝐵))
12	11	sseq1d 3925	. . . . 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 9693	. . . . . . 7 ⊢ (𝑎 ∈ V → ((𝑈‘𝑎)‘∅) = 𝑎)
15		tcid 9034	. . . . . . 7 ⊢ (𝑎 ∈ V → 𝑎 ⊆ (TC‘𝑎))
16	14, 15	eqsstrd 3932	. . . . . 6 ⊢ (𝑎 ∈ V → ((𝑈‘𝑎)‘∅) ⊆ (TC‘𝑎))
17	16	elv 3445	. . . . 5 ⊢ ((𝑈‘𝑎)‘∅) ⊆ (TC‘𝑎)
18	13	itunisuc 9694	. . . . . . 7 ⊢ ((𝑈‘𝑎)‘suc 𝑐) = ∪ ((𝑈‘𝑎)‘𝑐)
19		tctr 9035	. . . . . . . . . 10 ⊢ Tr (TC‘𝑎)
20		pwtr 5244	. . . . . . . . . 10 ⊢ (Tr (TC‘𝑎) ↔ Tr 𝒫 (TC‘𝑎))
21	19, 20	mpbi 231	. . . . . . . . 9 ⊢ Tr 𝒫 (TC‘𝑎)
22		trss 5079	. . . . . . . . 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 6558	. . . . . . . . 9 ⊢ ((𝑈‘𝑎)‘𝑐) ∈ V
25	24	elpw 4465	. . . . . . . 8 ⊢ (((𝑈‘𝑎)‘𝑐) ∈ 𝒫 (TC‘𝑎) ↔ ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎))
26		sspwuni 4927	. . . . . . . 8 ⊢ (((𝑈‘𝑎)‘𝑐) ⊆ 𝒫 (TC‘𝑎) ↔ ∪ ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎))
27	23, 25, 26	3imtr3i 292	. . . . . . 7 ⊢ (((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎) → ∪ ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎))
28	18, 27	eqsstrid 3942	. . . . . 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 7471	. . . 4 ⊢ (𝐵 ∈ ω → ((𝑈‘𝑎)‘𝐵) ⊆ (TC‘𝑎))
31		vex 3443	. . . . . . . 8 ⊢ 𝑎 ∈ V
32	13	itunifn 9692	. . . . . . . 8 ⊢ (𝑎 ∈ V → (𝑈‘𝑎) Fn ω)
33		fndm 6332	. . . . . . . 8 ⊢ ((𝑈‘𝑎) Fn ω → dom (𝑈‘𝑎) = ω)
34	31, 32, 33	mp2b 10	. . . . . . 7 ⊢ dom (𝑈‘𝑎) = ω
35	34	eleq2i 2876	. . . . . 6 ⊢ (𝐵 ∈ dom (𝑈‘𝑎) ↔ 𝐵 ∈ ω)
36		ndmfv 6575	. . . . . 6 ⊢ (¬ 𝐵 ∈ dom (𝑈‘𝑎) → ((𝑈‘𝑎)‘𝐵) = ∅)
37	35, 36	sylnbir 332	. . . . 5 ⊢ (¬ 𝐵 ∈ ω → ((𝑈‘𝑎)‘𝐵) = ∅)
38		0ss 4276	. . . . 5 ⊢ ∅ ⊆ (TC‘𝑎)
39	37, 38	syl6eqss 3948	. . . 4 ⊢ (¬ 𝐵 ∈ ω → ((𝑈‘𝑎)‘𝐵) ⊆ (TC‘𝑎))
40	30, 39	pm2.61i 183	. . 3 ⊢ ((𝑈‘𝑎)‘𝐵) ⊆ (TC‘𝑎)
41	4, 40	vtoclg 3513	. 2 ⊢ (𝐴 ∈ V → ((𝑈‘𝐴)‘𝐵) ⊆ (TC‘𝐴))
42		fv2prc 6585	. . 3 ⊢ (¬ 𝐴 ∈ V → ((𝑈‘𝐴)‘𝐵) = ∅)
43		0ss 4276	. . 3 ⊢ ∅ ⊆ (TC‘𝐴)
44	42, 43	syl6eqss 3948	. 2 ⊢ (¬ 𝐴 ∈ V → ((𝑈‘𝐴)‘𝐵) ⊆ (TC‘𝐴))
45	41, 44	pm2.61i 183	1 ⊢ ((𝑈‘𝐴)‘𝐵) ⊆ (TC‘𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1525   ∈ wcel 2083  Vcvv 3440   ⊆ wss 3865  ∅c0 4217  𝒫 cpw 4459  ∪ cuni 4751   ↦ cmpt 5047  Tr wtr 5070  dom cdm 5450   ↾ cres 5452  suc csuc 6075   Fn wfn 6227  ‘cfv 6232  ωcom 7443  reccrdg 7904  TCctc 9031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-inf2 8957

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-iin 4834  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-om 7444  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-tc 9032

This theorem is referenced by:  itunitc  9696