Theorem iunrelexpmin2 37023

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

Hypothesis

Ref	Expression
iunrelexpmin2.def	⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛))

Assertion

Ref	Expression
iunrelexpmin2	⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ0) → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 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 iunrelexpmin2

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

Step	Hyp	Ref	Expression
1		iunrelexpmin2.def	. . . . 5 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛))
2	1	a1i 11	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ0) → 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛)))
3		simplr 788	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ0) ∧ 𝑟 = 𝑅) → 𝑁 = ℕ0)
4		simpr 476	. . . . . 6 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ0) ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
5	4	oveq1d 6564	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ0) ∧ 𝑟 = 𝑅) → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛))
6	3, 5	iuneq12d 4482	. . . 4 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ0) ∧ 𝑟 = 𝑅) → ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
7		elex 3185	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
8	7	adantr 480	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ0) → 𝑅 ∈ V)
9		nn0ex 11175	. . . . . 6 ⊢ ℕ0 ∈ V
10		ovex 6577	. . . . . 6 ⊢ (𝑅↑𝑟𝑛) ∈ V
11	9, 10	iunex 7039	. . . . 5 ⊢ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ∈ V
12	11	a1i 11	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ0) → ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ∈ V)
13	2, 6, 8, 12	fvmptd 6197	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ0) → (𝐶‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
14		relexp0g 13610	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
15	14	sseq1d 3595	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟0) ⊆ 𝑠 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))
16		relexp1g 13614	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
17	16	sseq1d 3595	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟1) ⊆ 𝑠 ↔ 𝑅 ⊆ 𝑠))
18	15, 17	3anbi12d 1392	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)))
19		elnn0 11171	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ0 ↔ (𝑛 ∈ ℕ ∨ 𝑛 = 0))
20		oveq2 6557	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 1 → (𝑅↑𝑟𝑥) = (𝑅↑𝑟1))
21	20	sseq1d 3595	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → ((𝑅↑𝑟𝑥) ⊆ 𝑠 ↔ (𝑅↑𝑟1) ⊆ 𝑠))
22	21	imbi2d 329	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟𝑥) ⊆ 𝑠) ↔ ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟1) ⊆ 𝑠)))
23		oveq2 6557	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝑅↑𝑟𝑥) = (𝑅↑𝑟𝑦))
24	23	sseq1d 3595	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((𝑅↑𝑟𝑥) ⊆ 𝑠 ↔ (𝑅↑𝑟𝑦) ⊆ 𝑠))
25	24	imbi2d 329	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟𝑥) ⊆ 𝑠) ↔ ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟𝑦) ⊆ 𝑠)))
26		oveq2 6557	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑦 + 1) → (𝑅↑𝑟𝑥) = (𝑅↑𝑟(𝑦 + 1)))
27	26	sseq1d 3595	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 + 1) → ((𝑅↑𝑟𝑥) ⊆ 𝑠 ↔ (𝑅↑𝑟(𝑦 + 1)) ⊆ 𝑠))
28	27	imbi2d 329	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 + 1) → (((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟𝑥) ⊆ 𝑠) ↔ ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟(𝑦 + 1)) ⊆ 𝑠)))
29		oveq2 6557	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑛 → (𝑅↑𝑟𝑥) = (𝑅↑𝑟𝑛))
30	29	sseq1d 3595	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑛 → ((𝑅↑𝑟𝑥) ⊆ 𝑠 ↔ (𝑅↑𝑟𝑛) ⊆ 𝑠))
31	30	imbi2d 329	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑛 → (((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟𝑥) ⊆ 𝑠) ↔ ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟𝑛) ⊆ 𝑠)))
32		simpr2 1061	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟1) ⊆ 𝑠)
33		simp1 1054	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) ∧ (𝑅↑𝑟𝑦) ⊆ 𝑠) → 𝑦 ∈ ℕ)
34		1nn 10908	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℕ
35	34	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) ∧ (𝑅↑𝑟𝑦) ⊆ 𝑠) → 1 ∈ ℕ)
36		simp2l 1080	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) ∧ (𝑅↑𝑟𝑦) ⊆ 𝑠) → 𝑅 ∈ 𝑉)
37		relexpaddnn 13639	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ ∧ 1 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑦) ∘ (𝑅↑𝑟1)) = (𝑅↑𝑟(𝑦 + 1)))
38	33, 35, 36, 37	syl3anc 1318	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) ∧ (𝑅↑𝑟𝑦) ⊆ 𝑠) → ((𝑅↑𝑟𝑦) ∘ (𝑅↑𝑟1)) = (𝑅↑𝑟(𝑦 + 1)))
39		simp2r3 1158	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) ∧ (𝑅↑𝑟𝑦) ⊆ 𝑠) → (𝑠 ∘ 𝑠) ⊆ 𝑠)
40		simp3 1056	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) ∧ (𝑅↑𝑟𝑦) ⊆ 𝑠) → (𝑅↑𝑟𝑦) ⊆ 𝑠)
41		simp2r2 1157	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) ∧ (𝑅↑𝑟𝑦) ⊆ 𝑠) → (𝑅↑𝑟1) ⊆ 𝑠)
42	39, 40, 41	trrelssd 13560	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) ∧ (𝑅↑𝑟𝑦) ⊆ 𝑠) → ((𝑅↑𝑟𝑦) ∘ (𝑅↑𝑟1)) ⊆ 𝑠)
43	38, 42	eqsstr3d 3603	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℕ ∧ (𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) ∧ (𝑅↑𝑟𝑦) ⊆ 𝑠) → (𝑅↑𝑟(𝑦 + 1)) ⊆ 𝑠)
44	43	3exp 1256	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℕ → ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → ((𝑅↑𝑟𝑦) ⊆ 𝑠 → (𝑅↑𝑟(𝑦 + 1)) ⊆ 𝑠)))
45	44	a2d 29	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ → (((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟𝑦) ⊆ 𝑠) → ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟(𝑦 + 1)) ⊆ 𝑠)))
46	22, 25, 28, 31, 32, 45	nnind 10915	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟𝑛) ⊆ 𝑠))
47		simpr1 1060	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟0) ⊆ 𝑠)
48		oveq2 6557	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 0 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟0))
49	48	sseq1d 3595	. . . . . . . . . . . . 13 ⊢ (𝑛 = 0 → ((𝑅↑𝑟𝑛) ⊆ 𝑠 ↔ (𝑅↑𝑟0) ⊆ 𝑠))
50	47, 49	syl5ibr 235	. . . . . . . . . . . 12 ⊢ (𝑛 = 0 → ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟𝑛) ⊆ 𝑠))
51	46, 50	jaoi 393	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∨ 𝑛 = 0) → ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟𝑛) ⊆ 𝑠))
52	19, 51	sylbi 206	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0 → ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑅↑𝑟𝑛) ⊆ 𝑠))
53	52	com12 32	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → (𝑛 ∈ ℕ0 → (𝑅↑𝑟𝑛) ⊆ 𝑠))
54	53	ralrimiv 2948	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → ∀𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠)
55		iunss 4497	. . . . . . . 8 ⊢ (∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠 ↔ ∀𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠)
56	54, 55	sylibr 223	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ ((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) → ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠)
57	56	ex 449	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (((𝑅↑𝑟0) ⊆ 𝑠 ∧ (𝑅↑𝑟1) ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠))
58	18, 57	sylbird 249	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠))
59	58	adantr 480	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ0) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠))
60		sseq1 3589	. . . . 5 ⊢ ((𝐶‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) → ((𝐶‘𝑅) ⊆ 𝑠 ↔ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠))
61	60	imbi2d 329	. . . 4 ⊢ ((𝐶‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) → (((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (𝐶‘𝑅) ⊆ 𝑠) ↔ ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠)))
62	59, 61	syl5ibr 235	. . 3 ⊢ ((𝐶‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) → ((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ0) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (𝐶‘𝑅) ⊆ 𝑠)))
63	13, 62	mpcom 37	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ0) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (𝐶‘𝑅) ⊆ 𝑠))
64	63	alrimiv 1842	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ0) → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (𝐶‘𝑅) ⊆ 𝑠))

Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  ∪ ciun 4455   ↦ cmpt 4643   I cid 4948  dom cdm 5038  ran crn 5039   ↾ cres 5040   ∘ ccom 5042  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  ℕcn 10897  ℕ₀cn0 11169  ↑_𝑟crelexp 13608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-seq 12664  df-relexp 13609

This theorem is referenced by:  dfrtrcl3  37044