Theorem trclimalb2 43147

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 3489	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2	1	adantr 480	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → 𝑅 ∈ V)
3		oveq1 7422	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑘) = (𝑅↑𝑟𝑘))
4	3	iuneq2d 5021	. . . . . 6 ⊢ (𝑟 = 𝑅 → ∪ 𝑘 ∈ ℕ (𝑟↑𝑟𝑘) = ∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘))
5		dftrcl3 43141	. . . . . 6 ⊢ t+ = (𝑟 ∈ V ↦ ∪ 𝑘 ∈ ℕ (𝑟↑𝑟𝑘))
6		nnex 12243	. . . . . . 7 ⊢ ℕ ∈ V
7		ovex 7448	. . . . . . 7 ⊢ (𝑅↑𝑟𝑘) ∈ V
8	6, 7	iunex 7967	. . . . . 6 ⊢ ∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘) ∈ V
9	4, 5, 8	fvmpt 7000	. . . . 5 ⊢ (𝑅 ∈ V → (t+‘𝑅) = ∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘))
10	9	imaeq1d 6057	. . . 4 ⊢ (𝑅 ∈ V → ((t+‘𝑅) “ 𝐴) = (∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘) “ 𝐴))
11		imaiun1 43072	. . . 4 ⊢ (∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘) “ 𝐴) = ∪ 𝑘 ∈ ℕ ((𝑅↑𝑟𝑘) “ 𝐴)
12	10, 11	eqtrdi 2784	. . 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 7423	. . . . . . . . 9 ⊢ (𝑥 = 1 → (𝑅↑𝑟𝑥) = (𝑅↑𝑟1))
15	14	imaeq1d 6057	. . . . . . . 8 ⊢ (𝑥 = 1 → ((𝑅↑𝑟𝑥) “ 𝐴) = ((𝑅↑𝑟1) “ 𝐴))
16	15	sseq1d 4010	. . . . . . 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 7423	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑅↑𝑟𝑥) = (𝑅↑𝑟𝑦))
19	18	imaeq1d 6057	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑅↑𝑟𝑥) “ 𝐴) = ((𝑅↑𝑟𝑦) “ 𝐴))
20	19	sseq1d 4010	. . . . . . 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 7423	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 + 1) → (𝑅↑𝑟𝑥) = (𝑅↑𝑟(𝑦 + 1)))
23	22	imaeq1d 6057	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → ((𝑅↑𝑟𝑥) “ 𝐴) = ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴))
24	23	sseq1d 4010	. . . . . . 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 7423	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → (𝑅↑𝑟𝑥) = (𝑅↑𝑟𝑘))
27	26	imaeq1d 6057	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → ((𝑅↑𝑟𝑥) “ 𝐴) = ((𝑅↑𝑟𝑘) “ 𝐴))
28	27	sseq1d 4010	. . . . . . 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 15000	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
31	30	imaeq1d 6057	. . . . . . . 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 4169	. . . . . . . . 9 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
34		imass2 6101	. . . . . . . . 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 3989	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → (𝑅 “ 𝐴) ⊆ 𝐵)
38	32, 37	eqsstrd 4017	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((𝑅↑𝑟1) “ 𝐴) ⊆ 𝐵)
39		simp2l 1197	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → 𝑅 ∈ 𝑉)
40		simp1 1134	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → 𝑦 ∈ ℕ)
41		relexpsucnnl 15004	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ ℕ) → (𝑅↑𝑟(𝑦 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑦)))
42	41	imaeq1d 6057	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ ℕ) → ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴) = ((𝑅 ∘ (𝑅↑𝑟𝑦)) “ 𝐴))
43		imaco 6250	. . . . . . . . . . 11 ⊢ ((𝑅 ∘ (𝑅↑𝑟𝑦)) “ 𝐴) = (𝑅 “ ((𝑅↑𝑟𝑦) “ 𝐴))
44	42, 43	eqtrdi 2784	. . . . . . . . . 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 6101	. . . . . . . . . . 11 ⊢ (((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵 → (𝑅 “ ((𝑅↑𝑟𝑦) “ 𝐴)) ⊆ (𝑅 “ 𝐵))
47	46	3ad2ant3 1133	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ ((𝑅↑𝑟𝑦) “ 𝐴)) ⊆ (𝑅 “ 𝐵))
48		ssun2 4170	. . . . . . . . . . . 12 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
49		imass2 6101	. . . . . . . . . . . 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 1198	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵)
52	50, 51	sstrd 3989	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ 𝐵) ⊆ 𝐵)
53	47, 52	sstrd 3989	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ ((𝑅↑𝑟𝑦) “ 𝐴)) ⊆ 𝐵)
54	45, 53	eqsstrd 4017	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴) ⊆ 𝐵)
55	54	3exp 1117	. . . . . . 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 12255	. . . . 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 3141	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ∀𝑘 ∈ ℕ ((𝑅↑𝑟𝑘) “ 𝐴) ⊆ 𝐵)
60		iunss 5043	. . 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 4017	1 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((t+‘𝑅) “ 𝐴) ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  ∀wral 3057  Vcvv 3470   ∪ cun 3943   ⊆ wss 3945  ∪ ciun 4992   “ cima 5676   ∘ ccom 5677  ‘cfv 6543  (class class class)co 7415  1c1 11134   + caddc 11136  ℕcn 12237  t+ctcl 14959  ↑_𝑟crelexp 14993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7866  df-2nd 7989  df-frecs 8281  df-wrecs 8312  df-recs 8386  df-rdg 8425  df-er 8719  df-en 8959  df-dom 8960  df-sdom 8961  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-2 12300  df-n0 12498  df-z 12584  df-uz 12848  df-seq 13994  df-trcl 14961  df-relexp 14994

This theorem is referenced by:  brtrclfv2  43148  frege77d  43167