Theorem trclimalb2 43765

Description: Lower bound for image under a transitive closure. (Contributed by RP, 1-Jul-2020.)

Assertion

Ref	Expression
trclimalb2	⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((t+‘𝑅) “ 𝐴) ⊆ 𝐵)

Proof of Theorem trclimalb2

Dummy variables 𝑥 𝑘 𝑦 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3457	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2	1	adantr 480	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → 𝑅 ∈ V)
3		oveq1 7353	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑘) = (𝑅↑𝑟𝑘))
4	3	iuneq2d 4972	. . . . . 6 ⊢ (𝑟 = 𝑅 → ∪ 𝑘 ∈ ℕ (𝑟↑𝑟𝑘) = ∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘))
5		dftrcl3 43759	. . . . . 6 ⊢ t+ = (𝑟 ∈ V ↦ ∪ 𝑘 ∈ ℕ (𝑟↑𝑟𝑘))
6		nnex 12131	. . . . . . 7 ⊢ ℕ ∈ V
7		ovex 7379	. . . . . . 7 ⊢ (𝑅↑𝑟𝑘) ∈ V
8	6, 7	iunex 7900	. . . . . 6 ⊢ ∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘) ∈ V
9	4, 5, 8	fvmpt 6929	. . . . 5 ⊢ (𝑅 ∈ V → (t+‘𝑅) = ∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘))
10	9	imaeq1d 6008	. . . 4 ⊢ (𝑅 ∈ V → ((t+‘𝑅) “ 𝐴) = (∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘) “ 𝐴))
11		imaiun1 43690	. . . 4 ⊢ (∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘) “ 𝐴) = ∪ 𝑘 ∈ ℕ ((𝑅↑𝑟𝑘) “ 𝐴)
12	10, 11	eqtrdi 2782	. . 3 ⊢ (𝑅 ∈ V → ((t+‘𝑅) “ 𝐴) = ∪ 𝑘 ∈ ℕ ((𝑅↑𝑟𝑘) “ 𝐴))
13	2, 12	syl 17	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((t+‘𝑅) “ 𝐴) = ∪ 𝑘 ∈ ℕ ((𝑅↑𝑟𝑘) “ 𝐴))
14		oveq2 7354	. . . . . . . . 9 ⊢ (𝑥 = 1 → (𝑅↑𝑟𝑥) = (𝑅↑𝑟1))
15	14	imaeq1d 6008	. . . . . . . 8 ⊢ (𝑥 = 1 → ((𝑅↑𝑟𝑥) “ 𝐴) = ((𝑅↑𝑟1) “ 𝐴))
16	15	sseq1d 3966	. . . . . . 7 ⊢ (𝑥 = 1 → (((𝑅↑𝑟𝑥) “ 𝐴) ⊆ 𝐵 ↔ ((𝑅↑𝑟1) “ 𝐴) ⊆ 𝐵))
17	16	imbi2d 340	. . . . . 6 ⊢ (𝑥 = 1 → (((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((𝑅↑𝑟𝑥) “ 𝐴) ⊆ 𝐵) ↔ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((𝑅↑𝑟1) “ 𝐴) ⊆ 𝐵)))
18		oveq2 7354	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑅↑𝑟𝑥) = (𝑅↑𝑟𝑦))
19	18	imaeq1d 6008	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑅↑𝑟𝑥) “ 𝐴) = ((𝑅↑𝑟𝑦) “ 𝐴))
20	19	sseq1d 3966	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((𝑅↑𝑟𝑥) “ 𝐴) ⊆ 𝐵 ↔ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵))
21	20	imbi2d 340	. . . . . 6 ⊢ (𝑥 = 𝑦 → (((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((𝑅↑𝑟𝑥) “ 𝐴) ⊆ 𝐵) ↔ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵)))
22		oveq2 7354	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 + 1) → (𝑅↑𝑟𝑥) = (𝑅↑𝑟(𝑦 + 1)))
23	22	imaeq1d 6008	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → ((𝑅↑𝑟𝑥) “ 𝐴) = ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴))
24	23	sseq1d 3966	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (((𝑅↑𝑟𝑥) “ 𝐴) ⊆ 𝐵 ↔ ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴) ⊆ 𝐵))
25	24	imbi2d 340	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((𝑅↑𝑟𝑥) “ 𝐴) ⊆ 𝐵) ↔ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴) ⊆ 𝐵)))
26		oveq2 7354	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → (𝑅↑𝑟𝑥) = (𝑅↑𝑟𝑘))
27	26	imaeq1d 6008	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → ((𝑅↑𝑟𝑥) “ 𝐴) = ((𝑅↑𝑟𝑘) “ 𝐴))
28	27	sseq1d 3966	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (((𝑅↑𝑟𝑥) “ 𝐴) ⊆ 𝐵 ↔ ((𝑅↑𝑟𝑘) “ 𝐴) ⊆ 𝐵))
29	28	imbi2d 340	. . . . . 6 ⊢ (𝑥 = 𝑘 → (((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((𝑅↑𝑟𝑥) “ 𝐴) ⊆ 𝐵) ↔ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((𝑅↑𝑟𝑘) “ 𝐴) ⊆ 𝐵)))
30		relexp1g 14933	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
31	30	imaeq1d 6008	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟1) “ 𝐴) = (𝑅 “ 𝐴))
32	31	adantr 480	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((𝑅↑𝑟1) “ 𝐴) = (𝑅 “ 𝐴))
33		ssun1 4128	. . . . . . . . 9 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
34		imass2 6051	. . . . . . . . 9 ⊢ (𝐴 ⊆ (𝐴 ∪ 𝐵) → (𝑅 “ 𝐴) ⊆ (𝑅 “ (𝐴 ∪ 𝐵)))
35	33, 34	mp1i 13	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → (𝑅 “ 𝐴) ⊆ (𝑅 “ (𝐴 ∪ 𝐵)))
36		simpr 484	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵)
37	35, 36	sstrd 3945	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → (𝑅 “ 𝐴) ⊆ 𝐵)
38	32, 37	eqsstrd 3969	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((𝑅↑𝑟1) “ 𝐴) ⊆ 𝐵)
39		simp2l 1200	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → 𝑅 ∈ 𝑉)
40		simp1 1136	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → 𝑦 ∈ ℕ)
41		relexpsucnnl 14937	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ ℕ) → (𝑅↑𝑟(𝑦 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑦)))
42	41	imaeq1d 6008	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ ℕ) → ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴) = ((𝑅 ∘ (𝑅↑𝑟𝑦)) “ 𝐴))
43		imaco 6198	. . . . . . . . . . 11 ⊢ ((𝑅 ∘ (𝑅↑𝑟𝑦)) “ 𝐴) = (𝑅 “ ((𝑅↑𝑟𝑦) “ 𝐴))
44	42, 43	eqtrdi 2782	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ ℕ) → ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴) = (𝑅 “ ((𝑅↑𝑟𝑦) “ 𝐴)))
45	39, 40, 44	syl2anc 584	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴) = (𝑅 “ ((𝑅↑𝑟𝑦) “ 𝐴)))
46		imass2 6051	. . . . . . . . . . 11 ⊢ (((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵 → (𝑅 “ ((𝑅↑𝑟𝑦) “ 𝐴)) ⊆ (𝑅 “ 𝐵))
47	46	3ad2ant3 1135	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ ((𝑅↑𝑟𝑦) “ 𝐴)) ⊆ (𝑅 “ 𝐵))
48		ssun2 4129	. . . . . . . . . . . 12 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
49		imass2 6051	. . . . . . . . . . . 12 ⊢ (𝐵 ⊆ (𝐴 ∪ 𝐵) → (𝑅 “ 𝐵) ⊆ (𝑅 “ (𝐴 ∪ 𝐵)))
50	48, 49	mp1i 13	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ 𝐵) ⊆ (𝑅 “ (𝐴 ∪ 𝐵)))
51		simp2r 1201	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵)
52	50, 51	sstrd 3945	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ 𝐵) ⊆ 𝐵)
53	47, 52	sstrd 3945	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ ((𝑅↑𝑟𝑦) “ 𝐴)) ⊆ 𝐵)
54	45, 53	eqsstrd 3969	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴) ⊆ 𝐵)
55	54	3exp 1119	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → (((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵 → ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴) ⊆ 𝐵)))
56	55	a2d 29	. . . . . 6 ⊢ (𝑦 ∈ ℕ → (((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴) ⊆ 𝐵)))
57	17, 21, 25, 29, 38, 56	nnind 12143	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((𝑅↑𝑟𝑘) “ 𝐴) ⊆ 𝐵))
58	57	com12 32	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → (𝑘 ∈ ℕ → ((𝑅↑𝑟𝑘) “ 𝐴) ⊆ 𝐵))
59	58	ralrimiv 3123	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ∀𝑘 ∈ ℕ ((𝑅↑𝑟𝑘) “ 𝐴) ⊆ 𝐵)
60		iunss 4994	. . 3 ⊢ (∪ 𝑘 ∈ ℕ ((𝑅↑𝑟𝑘) “ 𝐴) ⊆ 𝐵 ↔ ∀𝑘 ∈ ℕ ((𝑅↑𝑟𝑘) “ 𝐴) ⊆ 𝐵)
61	59, 60	sylibr 234	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ∪ 𝑘 ∈ ℕ ((𝑅↑𝑟𝑘) “ 𝐴) ⊆ 𝐵)
62	13, 61	eqsstrd 3969	1 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((t+‘𝑅) “ 𝐴) ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ∪ cun 3900   ⊆ wss 3902  ∪ ciun 4941   “ cima 5619   ∘ ccom 5620  ‘cfv 6481  (class class class)co 7346  1c1 11007   + caddc 11009  ℕcn 12125  t+ctcl 14892  ↑_𝑟crelexp 14926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-n0 12382  df-z 12469  df-uz 12733  df-seq 13909  df-trcl 14894  df-relexp 14927

This theorem is referenced by:  brtrclfv2  43766  frege77d  43785