Theorem trclimalb2 44263

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 3474	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2	1	adantr 484	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → 𝑅 ∈ V)
3		oveq1 7398	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑘) = (𝑅↑𝑟𝑘))
4	3	iuneq2d 4977	. . . . . 6 ⊢ (𝑟 = 𝑅 → ∪ 𝑘 ∈ ℕ (𝑟↑𝑟𝑘) = ∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘))
5		dftrcl3 44257	. . . . . 6 ⊢ t+ = (𝑟 ∈ V ↦ ∪ 𝑘 ∈ ℕ (𝑟↑𝑟𝑘))
6		nnex 12210	. . . . . . 7 ⊢ ℕ ∈ V
7		ovex 7424	. . . . . . 7 ⊢ (𝑅↑𝑟𝑘) ∈ V
8	6, 7	iunex 7944	. . . . . 6 ⊢ ∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘) ∈ V
9	4, 5, 8	fvmpt 6970	. . . . 5 ⊢ (𝑅 ∈ V → (t+‘𝑅) = ∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘))
10	9	imaeq1d 6044	. . . 4 ⊢ (𝑅 ∈ V → ((t+‘𝑅) “ 𝐴) = (∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘) “ 𝐴))
11		imaiun1 44188	. . . 4 ⊢ (∪ 𝑘 ∈ ℕ (𝑅↑𝑟𝑘) “ 𝐴) = ∪ 𝑘 ∈ ℕ ((𝑅↑𝑟𝑘) “ 𝐴)
12	10, 11	eqtrdi 2812	. . 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 7399	. . . . . . . . 9 ⊢ (𝑥 = 1 → (𝑅↑𝑟𝑥) = (𝑅↑𝑟1))
15	14	imaeq1d 6044	. . . . . . . 8 ⊢ (𝑥 = 1 → ((𝑅↑𝑟𝑥) “ 𝐴) = ((𝑅↑𝑟1) “ 𝐴))
16	15	sseq1d 3965	. . . . . . 7 ⊢ (𝑥 = 1 → (((𝑅↑𝑟𝑥) “ 𝐴) ⊆ 𝐵 ↔ ((𝑅↑𝑟1) “ 𝐴) ⊆ 𝐵))
17	16	imbi2d 342	. . . . . 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 7399	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑅↑𝑟𝑥) = (𝑅↑𝑟𝑦))
19	18	imaeq1d 6044	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑅↑𝑟𝑥) “ 𝐴) = ((𝑅↑𝑟𝑦) “ 𝐴))
20	19	sseq1d 3965	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((𝑅↑𝑟𝑥) “ 𝐴) ⊆ 𝐵 ↔ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵))
21	20	imbi2d 342	. . . . . 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 7399	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 + 1) → (𝑅↑𝑟𝑥) = (𝑅↑𝑟(𝑦 + 1)))
23	22	imaeq1d 6044	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → ((𝑅↑𝑟𝑥) “ 𝐴) = ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴))
24	23	sseq1d 3965	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (((𝑅↑𝑟𝑥) “ 𝐴) ⊆ 𝐵 ↔ ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴) ⊆ 𝐵))
25	24	imbi2d 342	. . . . . 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 7399	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → (𝑅↑𝑟𝑥) = (𝑅↑𝑟𝑘))
27	26	imaeq1d 6044	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → ((𝑅↑𝑟𝑥) “ 𝐴) = ((𝑅↑𝑟𝑘) “ 𝐴))
28	27	sseq1d 3965	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (((𝑅↑𝑟𝑥) “ 𝐴) ⊆ 𝐵 ↔ ((𝑅↑𝑟𝑘) “ 𝐴) ⊆ 𝐵))
29	28	imbi2d 342	. . . . . 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 15033	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
31	30	imaeq1d 6044	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟1) “ 𝐴) = (𝑅 “ 𝐴))
32	31	adantr 484	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((𝑅↑𝑟1) “ 𝐴) = (𝑅 “ 𝐴))
33		ssun1 4128	. . . . . . . . 9 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
34		imass2 6087	. . . . . . . . 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 488	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵)
37	35, 36	sstrd 3944	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → (𝑅 “ 𝐴) ⊆ 𝐵)
38	32, 37	eqsstrd 3968	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((𝑅↑𝑟1) “ 𝐴) ⊆ 𝐵)
39		simp2l 1212	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → 𝑅 ∈ 𝑉)
40		simp1 1148	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → 𝑦 ∈ ℕ)
41		relexpsucnnl 15037	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ ℕ) → (𝑅↑𝑟(𝑦 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑦)))
42	41	imaeq1d 6044	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ ℕ) → ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴) = ((𝑅 ∘ (𝑅↑𝑟𝑦)) “ 𝐴))
43		imaco 6233	. . . . . . . . . . 11 ⊢ ((𝑅 ∘ (𝑅↑𝑟𝑦)) “ 𝐴) = (𝑅 “ ((𝑅↑𝑟𝑦) “ 𝐴))
44	42, 43	eqtrdi 2812	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ ℕ) → ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴) = (𝑅 “ ((𝑅↑𝑟𝑦) “ 𝐴)))
45	39, 40, 44	syl2anc 593	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴) = (𝑅 “ ((𝑅↑𝑟𝑦) “ 𝐴)))
46		imass2 6087	. . . . . . . . . . 11 ⊢ (((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵 → (𝑅 “ ((𝑅↑𝑟𝑦) “ 𝐴)) ⊆ (𝑅 “ 𝐵))
47	46	3ad2ant3 1147	. . . . . . . . . 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 6087	. . . . . . . . . . . 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 1213	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵)
52	50, 51	sstrd 3944	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ 𝐵) ⊆ 𝐵)
53	47, 52	sstrd 3944	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ ((𝑅↑𝑟𝑦) “ 𝐴)) ⊆ 𝐵)
54	45, 53	eqsstrd 3968	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) ∧ ((𝑅↑𝑟𝑦) “ 𝐴) ⊆ 𝐵) → ((𝑅↑𝑟(𝑦 + 1)) “ 𝐴) ⊆ 𝐵)
55	54	3exp 1131	. . . . . . 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 12222	. . . . 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 3152	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ∀𝑘 ∈ ℕ ((𝑅↑𝑟𝑘) “ 𝐴) ⊆ 𝐵)
60		iunss 4999	. . 3 ⊢ (∪ 𝑘 ∈ ℕ ((𝑅↑𝑟𝑘) “ 𝐴) ⊆ 𝐵 ↔ ∀𝑘 ∈ ℕ ((𝑅↑𝑟𝑘) “ 𝐴) ⊆ 𝐵)
61	59, 60	sylibr 236	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ∪ 𝑘 ∈ ℕ ((𝑅↑𝑟𝑘) “ 𝐴) ⊆ 𝐵)
62	13, 61	eqsstrd 3968	1 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((t+‘𝑅) “ 𝐴) ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075  Vcvv 3453   ∪ cun 3900   ⊆ wss 3902  ∪ ciun 4946   “ cima 5646   ∘ ccom 5647  ‘cfv 6516  (class class class)co 7391  1c1 11068   + caddc 11070  ℕcn 12204  t+ctcl 14992  ↑_𝑟crelexp 15026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-cnex 11123  ax-resscn 11124  ax-1cn 11125  ax-icn 11126  ax-addcl 11127  ax-addrcl 11128  ax-mulcl 11129  ax-mulrcl 11130  ax-mulcom 11131  ax-addass 11132  ax-mulass 11133  ax-distr 11134  ax-i2m1 11135  ax-1ne0 11136  ax-1rid 11137  ax-rnegex 11138  ax-rrecex 11139  ax-cnre 11140  ax-pre-lttri 11141  ax-pre-lttrn 11142  ax-pre-ltadd 11143  ax-pre-mulgt0 11144

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-om 7842  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-er 8672  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11212  df-mnf 11213  df-xr 11214  df-ltxr 11215  df-le 11216  df-sub 11410  df-neg 11411  df-nn 12205  df-2 12274  df-n0 12476  df-z 12563  df-uz 12834  df-seq 14009  df-trcl 14994  df-relexp 15027

This theorem is referenced by:  brtrclfv2  44264  frege77d  44283