Theorem iunrelexpmin1 38000

Description: The indexed union of relation exponentiation over the natural numbers is the minimum transitive relation that includes the relation. (Contributed by RP, 4-Jun-2020.)

Hypothesis

Ref	Expression
iunrelexpmin1.def	⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛))

Assertion

Ref	Expression
iunrelexpmin1	⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ) → ∀𝑠((𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (𝐶‘𝑅) ⊆ 𝑠))

Distinct variable groups:   𝑛,𝑟,𝐶,𝑁   𝑁,𝑠   𝑅,𝑛,𝑟   𝑅,𝑠   𝑛,𝑉,𝑟   𝑉,𝑠,𝑛

Allowed substitution hint:   𝐶(𝑠)

Proof of Theorem iunrelexpmin1

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

Step	Hyp	Ref	Expression
1		iunrelexpmin1.def	. . . . 5 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛))
2	1	a1i 11	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ) → 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛)))
3		simplr 792	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ) ∧ 𝑟 = 𝑅) → 𝑁 = ℕ)
4		simpr 477	. . . . . 6 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ) ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
5	4	oveq1d 6665	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ) ∧ 𝑟 = 𝑅) → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛))
6	3, 5	iuneq12d 4546	. . . 4 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ) ∧ 𝑟 = 𝑅) → ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛))
7		elex 3212	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
8	7	adantr 481	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ) → 𝑅 ∈ V)
9		nnex 11026	. . . . . 6 ⊢ ℕ ∈ V
10		ovex 6678	. . . . . 6 ⊢ (𝑅↑𝑟𝑛) ∈ V
11	9, 10	iunex 7147	. . . . 5 ⊢ ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) ∈ V
12	11	a1i 11	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ) → ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) ∈ V)
13	2, 6, 8, 12	fvmptd 6288	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ) → (𝐶‘𝑅) = ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛))
14		relexp1g 13766	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
15	14	sseq1d 3632	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟1) ⊆ 𝑠 ↔ 𝑅 ⊆ 𝑠))
16	15	anbi1d 741	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) ↔ (𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)))
17		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (𝑅↑𝑟𝑥) = (𝑅↑𝑟1))
18	17	sseq1d 3632	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → ((𝑅↑𝑟𝑥) ⊆ 𝑠 ↔ (𝑅↑𝑟1) ⊆ 𝑠))
19	18	imbi2d 330	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟𝑥) ⊆ 𝑠) ↔ ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟1) ⊆ 𝑠)))
20		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑅↑𝑟𝑥) = (𝑅↑𝑟𝑦))
21	20	sseq1d 3632	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑅↑𝑟𝑥) ⊆ 𝑠 ↔ (𝑅↑𝑟𝑦) ⊆ 𝑠))
22	21	imbi2d 330	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟𝑥) ⊆ 𝑠) ↔ ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟𝑦) ⊆ 𝑠)))
23		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 + 1) → (𝑅↑𝑟𝑥) = (𝑅↑𝑟(𝑦 + 1)))
24	23	sseq1d 3632	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦 + 1) → ((𝑅↑𝑟𝑥) ⊆ 𝑠 ↔ (𝑅↑𝑟(𝑦 + 1)) ⊆ 𝑠))
25	24	imbi2d 330	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 + 1) → (((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟𝑥) ⊆ 𝑠) ↔ ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟(𝑦 + 1)) ⊆ 𝑠)))
26		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑛 → (𝑅↑𝑟𝑥) = (𝑅↑𝑟𝑛))
27	26	sseq1d 3632	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑛 → ((𝑅↑𝑟𝑥) ⊆ 𝑠 ↔ (𝑅↑𝑟𝑛) ⊆ 𝑠))
28	27	imbi2d 330	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑛 → (((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟𝑥) ⊆ 𝑠) ↔ ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟𝑛) ⊆ 𝑠)))
29		simprl 794	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟1) ⊆ 𝑠)
30		simp1 1061	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) ∧ (𝑅↑𝑟𝑦) ⊆ 𝑠) → 𝑦 ∈ ℕ)
31		1nn 11031	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ
32	31	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) ∧ (𝑅↑𝑟𝑦) ⊆ 𝑠) → 1 ∈ ℕ)
33		simp2l 1087	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) ∧ (𝑅↑𝑟𝑦) ⊆ 𝑠) → 𝑅 ∈ 𝑉)
34		relexpaddnn 13791	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℕ ∧ 1 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑦) ∘ (𝑅↑𝑟1)) = (𝑅↑𝑟(𝑦 + 1)))
35	30, 32, 33, 34	syl3anc 1326	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) ∧ (𝑅↑𝑟𝑦) ⊆ 𝑠) → ((𝑅↑𝑟𝑦) ∘ (𝑅↑𝑟1)) = (𝑅↑𝑟(𝑦 + 1)))
36		simp2rr 1131	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) ∧ (𝑅↑𝑟𝑦) ⊆ 𝑠) → (𝑠 ∘ 𝑠) ⊆ 𝑠)
37		simp3 1063	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) ∧ (𝑅↑𝑟𝑦) ⊆ 𝑠) → (𝑅↑𝑟𝑦) ⊆ 𝑠)
38		simp2rl 1130	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) ∧ (𝑅↑𝑟𝑦) ⊆ 𝑠) → (𝑅↑𝑟1) ⊆ 𝑠)
39	36, 37, 38	trrelssd 13712	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) ∧ (𝑅↑𝑟𝑦) ⊆ 𝑠) → ((𝑅↑𝑟𝑦) ∘ (𝑅↑𝑟1)) ⊆ 𝑠)
40	35, 39	eqsstr3d 3640	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) ∧ (𝑅↑𝑟𝑦) ⊆ 𝑠) → (𝑅↑𝑟(𝑦 + 1)) ⊆ 𝑠)
41	40	3exp 1264	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ → ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → ((𝑅↑𝑟𝑦) ⊆ 𝑠 → (𝑅↑𝑟(𝑦 + 1)) ⊆ 𝑠)))
42	41	a2d 29	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → (((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟𝑦) ⊆ 𝑠) → ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟(𝑦 + 1)) ⊆ 𝑠)))
43	19, 22, 25, 28, 29, 42	nnind 11038	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟𝑛) ⊆ 𝑠))
44	43	com12 32	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑛 ∈ ℕ → (𝑅↑𝑟𝑛) ⊆ 𝑠))
45	44	ralrimiv 2965	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → ∀𝑛 ∈ ℕ (𝑅↑𝑟𝑛) ⊆ 𝑠)
46		iunss 4561	. . . . . . . 8 ⊢ (∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) ⊆ 𝑠 ↔ ∀𝑛 ∈ ℕ (𝑅↑𝑟𝑛) ⊆ 𝑠)
47	45, 46	sylibr 224	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) ⊆ 𝑠)
48	47	ex 450	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (((𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) ⊆ 𝑠))
49	16, 48	sylbird 250	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → ((𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) ⊆ 𝑠))
50	49	adantr 481	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ) → ((𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) ⊆ 𝑠))
51		sseq1 3626	. . . . 5 ⊢ ((𝐶‘𝑅) = ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) → ((𝐶‘𝑅) ⊆ 𝑠 ↔ ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) ⊆ 𝑠))
52	51	imbi2d 330	. . . 4 ⊢ ((𝐶‘𝑅) = ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) → (((𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (𝐶‘𝑅) ⊆ 𝑠) ↔ ((𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) ⊆ 𝑠)))
53	50, 52	syl5ibr 236	. . 3 ⊢ ((𝐶‘𝑅) = ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) → ((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ) → ((𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (𝐶‘𝑅) ⊆ 𝑠)))
54	13, 53	mpcom 38	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ) → ((𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (𝐶‘𝑅) ⊆ 𝑠))
55	54	alrimiv 1855	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ) → ∀𝑠((𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (𝐶‘𝑅) ⊆ 𝑠))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  ∪ ciun 4520   ↦ cmpt 4729   ∘ ccom 5118  ‘cfv 5888  (class class class)co 6650  1c1 9937   + caddc 9939  ℕcn 11020  ↑_𝑟crelexp 13760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-relexp 13761

This theorem is referenced by:  dftrcl3  38012