Theorem trclimalb2 43688

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 3509	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2	1	adantr 480	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → 𝑅 ∈ V)
3		oveq1 7455	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑘) = (𝑅↑𝑟𝑘))
4	3	iuneq2d 5045	. . . . . 6 ⊢ (𝑟 = 𝑅 → ∪ 𝑘 ∈ ℕ (𝑟↑𝑟𝑘) = ∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘))
5		dftrcl3 43682	. . . . . 6 ⊢ t+ = (𝑟 ∈ V ↦ ∪ 𝑘 ∈ ℕ (𝑟↑𝑟𝑘))
6		nnex 12299	. . . . . . 7 ⊢ ℕ ∈ V
7		ovex 7481	. . . . . . 7 ⊢ (𝑅↑𝑟𝑘) ∈ V
8	6, 7	iunex 8009	. . . . . 6 ⊢ ∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘) ∈ V
9	4, 5, 8	fvmpt 7029	. . . . 5 ⊢ (𝑅 ∈ V → (t+‘𝑅) = ∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘))
10	9	imaeq1d 6088	. . . 4 ⊢ (𝑅 ∈ V → ((t+‘𝑅) “ 𝐴) = (∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘) “ 𝐴))
11		imaiun1 43613	. . . 4 ⊢ (∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘) “ 𝐴) = ∪ 𝑘 ∈ ℕ ((𝑅↑𝑟𝑘) “ 𝐴)
12	10, 11	eqtrdi 2796	. . 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 7456	. . . . . . . . 9 ⊢ (𝑥 = 1 → (𝑅↑𝑟𝑥) = (𝑅↑𝑟1))
15	14	imaeq1d 6088	. . . . . . . 8 ⊢ (𝑥 = 1 → ((𝑅↑𝑟𝑥) “ 𝐴) = ((𝑅↑𝑟1) “ 𝐴))
16	15	sseq1d 4040	. . . . . . 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 7456	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑅↑𝑟𝑥) = (𝑅↑𝑟𝑦))
19	18	imaeq1d 6088	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑅↑𝑟𝑥) “ 𝐴) = ((𝑅↑𝑟𝑦) “ 𝐴))
20	19	sseq1d 4040	. . . . . . 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 7456	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 + 1) → (𝑅↑𝑟𝑥) = (𝑅↑𝑟(𝑦 + 1)))
23	22	imaeq1d 6088	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → ((𝑅↑𝑟𝑥) “ 𝐴) = ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴))
24	23	sseq1d 4040	. . . . . . 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 7456	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → (𝑅↑𝑟𝑥) = (𝑅↑𝑟𝑘))
27	26	imaeq1d 6088	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → ((𝑅↑𝑟𝑥) “ 𝐴) = ((𝑅↑𝑟𝑘) “ 𝐴))
28	27	sseq1d 4040	. . . . . . 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 15075	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
31	30	imaeq1d 6088	. . . . . . . 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 4201	. . . . . . . . 9 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
34		imass2 6132	. . . . . . . . 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 4019	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → (𝑅 “ 𝐴) ⊆ 𝐵)
38	32, 37	eqsstrd 4047	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((𝑅↑𝑟1) “ 𝐴) ⊆ 𝐵)
39		simp2l 1199	. . . . . . . . . 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 15079	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ ℕ) → (𝑅↑𝑟(𝑦 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑦)))
42	41	imaeq1d 6088	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ ℕ) → ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴) = ((𝑅 ∘ (𝑅↑𝑟𝑦)) “ 𝐴))
43		imaco 6282	. . . . . . . . . . 11 ⊢ ((𝑅 ∘ (𝑅↑𝑟𝑦)) “ 𝐴) = (𝑅 “ ((𝑅↑𝑟𝑦) “ 𝐴))
44	42, 43	eqtrdi 2796	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ ℕ) → ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴) = (𝑅 “ ((𝑅↑𝑟𝑦) “ 𝐴)))
45	39, 40, 44	syl2anc 583	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴) = (𝑅 “ ((𝑅↑𝑟𝑦) “ 𝐴)))
46		imass2 6132	. . . . . . . . . . 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 4202	. . . . . . . . . . . 12 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
49		imass2 6132	. . . . . . . . . . . 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 1200	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵)
52	50, 51	sstrd 4019	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ 𝐵) ⊆ 𝐵)
53	47, 52	sstrd 4019	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ ((𝑅↑𝑟𝑦) “ 𝐴)) ⊆ 𝐵)
54	45, 53	eqsstrd 4047	. . . . . . . 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 12311	. . . . 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 3151	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ∀𝑘 ∈ ℕ ((𝑅↑𝑟𝑘) “ 𝐴) ⊆ 𝐵)
60		iunss 5068	. . 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 4047	1 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((t+‘𝑅) “ 𝐴) ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   ∪ cun 3974   ⊆ wss 3976  ∪ ciun 5015   “ cima 5703   ∘ ccom 5704  ‘cfv 6573  (class class class)co 7448  1c1 11185   + caddc 11187  ℕcn 12293  t+ctcl 15034  ↑_𝑟crelexp 15068

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-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

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-nel 3053  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-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-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-n0 12554  df-z 12640  df-uz 12904  df-seq 14053  df-trcl 15036  df-relexp 15069

This theorem is referenced by:  brtrclfv2  43689  frege77d  43708