Theorem trclimalb2 41334

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 3450	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2	1	adantr 481	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → 𝑅 ∈ V)
3		oveq1 7282	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑘) = (𝑅↑𝑟𝑘))
4	3	iuneq2d 4953	. . . . . 6 ⊢ (𝑟 = 𝑅 → ∪ 𝑘 ∈ ℕ (𝑟↑𝑟𝑘) = ∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘))
5		dftrcl3 41328	. . . . . 6 ⊢ t+ = (𝑟 ∈ V ↦ ∪ 𝑘 ∈ ℕ (𝑟↑𝑟𝑘))
6		nnex 11979	. . . . . . 7 ⊢ ℕ ∈ V
7		ovex 7308	. . . . . . 7 ⊢ (𝑅↑𝑟𝑘) ∈ V
8	6, 7	iunex 7811	. . . . . 6 ⊢ ∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘) ∈ V
9	4, 5, 8	fvmpt 6875	. . . . 5 ⊢ (𝑅 ∈ V → (t+‘𝑅) = ∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘))
10	9	imaeq1d 5968	. . . 4 ⊢ (𝑅 ∈ V → ((t+‘𝑅) “ 𝐴) = (∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘) “ 𝐴))
11		imaiun1 41259	. . . 4 ⊢ (∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘) “ 𝐴) = ∪ 𝑘 ∈ ℕ ((𝑅↑𝑟𝑘) “ 𝐴)
12	10, 11	eqtrdi 2794	. . 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 7283	. . . . . . . . 9 ⊢ (𝑥 = 1 → (𝑅↑𝑟𝑥) = (𝑅↑𝑟1))
15	14	imaeq1d 5968	. . . . . . . 8 ⊢ (𝑥 = 1 → ((𝑅↑𝑟𝑥) “ 𝐴) = ((𝑅↑𝑟1) “ 𝐴))
16	15	sseq1d 3952	. . . . . . 7 ⊢ (𝑥 = 1 → (((𝑅↑𝑟𝑥) “ 𝐴) ⊆ 𝐵 ↔ ((𝑅↑𝑟1) “ 𝐴) ⊆ 𝐵))
17	16	imbi2d 341	. . . . . 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 7283	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑅↑𝑟𝑥) = (𝑅↑𝑟𝑦))
19	18	imaeq1d 5968	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑅↑𝑟𝑥) “ 𝐴) = ((𝑅↑𝑟𝑦) “ 𝐴))
20	19	sseq1d 3952	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((𝑅↑𝑟𝑥) “ 𝐴) ⊆ 𝐵 ↔ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵))
21	20	imbi2d 341	. . . . . 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 7283	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 + 1) → (𝑅↑𝑟𝑥) = (𝑅↑𝑟(𝑦 + 1)))
23	22	imaeq1d 5968	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → ((𝑅↑𝑟𝑥) “ 𝐴) = ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴))
24	23	sseq1d 3952	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (((𝑅↑𝑟𝑥) “ 𝐴) ⊆ 𝐵 ↔ ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴) ⊆ 𝐵))
25	24	imbi2d 341	. . . . . 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 7283	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → (𝑅↑𝑟𝑥) = (𝑅↑𝑟𝑘))
27	26	imaeq1d 5968	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → ((𝑅↑𝑟𝑥) “ 𝐴) = ((𝑅↑𝑟𝑘) “ 𝐴))
28	27	sseq1d 3952	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (((𝑅↑𝑟𝑥) “ 𝐴) ⊆ 𝐵 ↔ ((𝑅↑𝑟𝑘) “ 𝐴) ⊆ 𝐵))
29	28	imbi2d 341	. . . . . 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 14737	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
31	30	imaeq1d 5968	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟1) “ 𝐴) = (𝑅 “ 𝐴))
32	31	adantr 481	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((𝑅↑𝑟1) “ 𝐴) = (𝑅 “ 𝐴))
33		ssun1 4106	. . . . . . . . 9 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
34		imass2 6010	. . . . . . . . 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 485	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵)
37	35, 36	sstrd 3931	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → (𝑅 “ 𝐴) ⊆ 𝐵)
38	32, 37	eqsstrd 3959	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((𝑅↑𝑟1) “ 𝐴) ⊆ 𝐵)
39		simp2l 1198	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → 𝑅 ∈ 𝑉)
40		simp1 1135	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → 𝑦 ∈ ℕ)
41		relexpsucnnl 14741	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ ℕ) → (𝑅↑𝑟(𝑦 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑦)))
42	41	imaeq1d 5968	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ ℕ) → ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴) = ((𝑅 ∘ (𝑅↑𝑟𝑦)) “ 𝐴))
43		imaco 6155	. . . . . . . . . . 11 ⊢ ((𝑅 ∘ (𝑅↑𝑟𝑦)) “ 𝐴) = (𝑅 “ ((𝑅↑𝑟𝑦) “ 𝐴))
44	42, 43	eqtrdi 2794	. . . . . . . . . 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 6010	. . . . . . . . . . 11 ⊢ (((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵 → (𝑅 “ ((𝑅↑𝑟𝑦) “ 𝐴)) ⊆ (𝑅 “ 𝐵))
47	46	3ad2ant3 1134	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ ((𝑅↑𝑟𝑦) “ 𝐴)) ⊆ (𝑅 “ 𝐵))
48		ssun2 4107	. . . . . . . . . . . 12 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
49		imass2 6010	. . . . . . . . . . . 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 1199	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵)
52	50, 51	sstrd 3931	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ 𝐵) ⊆ 𝐵)
53	47, 52	sstrd 3931	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ ((𝑅↑𝑟𝑦) “ 𝐴)) ⊆ 𝐵)
54	45, 53	eqsstrd 3959	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴) ⊆ 𝐵)
55	54	3exp 1118	. . . . . . 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 11991	. . . . 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 3102	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ∀𝑘 ∈ ℕ ((𝑅↑𝑟𝑘) “ 𝐴) ⊆ 𝐵)
60		iunss 4975	. . 3 ⊢ (∪ 𝑘 ∈ ℕ ((𝑅↑𝑟𝑘) “ 𝐴) ⊆ 𝐵 ↔ ∀𝑘 ∈ ℕ ((𝑅↑𝑟𝑘) “ 𝐴) ⊆ 𝐵)
61	59, 60	sylibr 233	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ∪ 𝑘 ∈ ℕ ((𝑅↑𝑟𝑘) “ 𝐴) ⊆ 𝐵)
62	13, 61	eqsstrd 3959	1 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((t+‘𝑅) “ 𝐴) ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ∪ cun 3885   ⊆ wss 3887  ∪ ciun 4924   “ cima 5592   ∘ ccom 5593  ‘cfv 6433  (class class class)co 7275  1c1 10872   + caddc 10874  ℕcn 11973  t+ctcl 14696  ↑_𝑟crelexp 14730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-seq 13722  df-trcl 14698  df-relexp 14731

This theorem is referenced by:  brtrclfv2  41335  frege77d  41354