Theorem trclun 15053

Description: Transitive closure of a union of relations. (Contributed by RP, 5-May-2020.)

Assertion

Ref	Expression
trclun	⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (t+‘(𝑅 ∪ 𝑆)) = (t+‘((t+‘𝑅) ∪ (t+‘𝑆))))

Proof of Theorem trclun

Dummy variables 𝑥 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		unss 4190	. . . . . . . . . 10 ⊢ ((𝑅 ⊆ 𝑥 ∧ 𝑆 ⊆ 𝑥) ↔ (𝑅 ∪ 𝑆) ⊆ 𝑥)
2		simpl 482	. . . . . . . . . 10 ⊢ ((𝑅 ⊆ 𝑥 ∧ 𝑆 ⊆ 𝑥) → 𝑅 ⊆ 𝑥)
3	1, 2	sylbir 235	. . . . . . . . 9 ⊢ ((𝑅 ∪ 𝑆) ⊆ 𝑥 → 𝑅 ⊆ 𝑥)
4		vex 3484	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
5		trcleq2lem 15030	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑥 → ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ↔ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)))
6	4, 5	elab 3679	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ↔ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥))
7	6	biimpri 228	. . . . . . . . 9 ⊢ ((𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) → 𝑥 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)})
8	3, 7	sylan 580	. . . . . . . 8 ⊢ (((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) → 𝑥 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)})
9		intss1 4963	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ⊆ 𝑥)
10	8, 9	syl 17	. . . . . . 7 ⊢ (((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ⊆ 𝑥)
11		simpr 484	. . . . . . . . . 10 ⊢ ((𝑅 ⊆ 𝑥 ∧ 𝑆 ⊆ 𝑥) → 𝑆 ⊆ 𝑥)
12	1, 11	sylbir 235	. . . . . . . . 9 ⊢ ((𝑅 ∪ 𝑆) ⊆ 𝑥 → 𝑆 ⊆ 𝑥)
13		trcleq2lem 15030	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑥 → ((𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) ↔ (𝑆 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)))
14	4, 13	elab 3679	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)} ↔ (𝑆 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥))
15	14	biimpri 228	. . . . . . . . 9 ⊢ ((𝑆 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) → 𝑥 ∈ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)})
16	12, 15	sylan 580	. . . . . . . 8 ⊢ (((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) → 𝑥 ∈ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)})
17		intss1 4963	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)} → ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)} ⊆ 𝑥)
18	16, 17	syl 17	. . . . . . 7 ⊢ (((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) → ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)} ⊆ 𝑥)
19	10, 18	unssd 4192	. . . . . 6 ⊢ (((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) → (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}) ⊆ 𝑥)
20		simpr 484	. . . . . 6 ⊢ (((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) → (𝑥 ∘ 𝑥) ⊆ 𝑥)
21	19, 20	jca 511	. . . . 5 ⊢ (((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) → ((∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥))
22		ssmin 4967	. . . . . . . 8 ⊢ 𝑅 ⊆ ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}
23		ssmin 4967	. . . . . . . 8 ⊢ 𝑆 ⊆ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}
24		unss12 4188	. . . . . . . 8 ⊢ ((𝑅 ⊆ ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∧ 𝑆 ⊆ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}) → (𝑅 ∪ 𝑆) ⊆ (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}))
25	22, 23, 24	mp2an 692	. . . . . . 7 ⊢ (𝑅 ∪ 𝑆) ⊆ (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)})
26		sstr 3992	. . . . . . 7 ⊢ (((𝑅 ∪ 𝑆) ⊆ (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}) ∧ (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}) ⊆ 𝑥) → (𝑅 ∪ 𝑆) ⊆ 𝑥)
27	25, 26	mpan 690	. . . . . 6 ⊢ ((∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}) ⊆ 𝑥 → (𝑅 ∪ 𝑆) ⊆ 𝑥)
28	27	anim1i 615	. . . . 5 ⊢ (((∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) → ((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥))
29	21, 28	impbii 209	. . . 4 ⊢ (((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) ↔ ((∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥))
30	29	abbii 2809	. . 3 ⊢ {𝑥 ∣ ((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = {𝑥 ∣ ((∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)}
31	30	inteqi 4950	. 2 ⊢ ∩ {𝑥 ∣ ((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = ∩ {𝑥 ∣ ((∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)}
32		unexg 7763	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ∪ 𝑆) ∈ V)
33		trclfv 15039	. . 3 ⊢ ((𝑅 ∪ 𝑆) ∈ V → (t+‘(𝑅 ∪ 𝑆)) = ∩ {𝑥 ∣ ((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
34	32, 33	syl 17	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (t+‘(𝑅 ∪ 𝑆)) = ∩ {𝑥 ∣ ((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
35		simpl 482	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑅 ∈ 𝑉)
36		trclfv 15039	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)})
37	35, 36	syl 17	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (t+‘𝑅) = ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)})
38		simpr 484	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑆 ∈ 𝑊)
39		trclfv 15039	. . . . . 6 ⊢ (𝑆 ∈ 𝑊 → (t+‘𝑆) = ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)})
40	38, 39	syl 17	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (t+‘𝑆) = ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)})
41	37, 40	uneq12d 4169	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((t+‘𝑅) ∪ (t+‘𝑆)) = (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}))
42	41	fveq2d 6910	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (t+‘((t+‘𝑅) ∪ (t+‘𝑆))) = (t+‘(∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)})))
43		fvex 6919	. . . . . 6 ⊢ (t+‘𝑅) ∈ V
44	36, 43	eqeltrrdi 2850	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∈ V)
45		fvex 6919	. . . . . 6 ⊢ (t+‘𝑆) ∈ V
46	39, 45	eqeltrrdi 2850	. . . . 5 ⊢ (𝑆 ∈ 𝑊 → ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)} ∈ V)
47		unexg 7763	. . . . 5 ⊢ ((∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∈ V ∧ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)} ∈ V) → (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}) ∈ V)
48	44, 46, 47	syl2an 596	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}) ∈ V)
49		trclfv 15039	. . . 4 ⊢ ((∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}) ∈ V → (t+‘(∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)})) = ∩ {𝑥 ∣ ((∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
50	48, 49	syl 17	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (t+‘(∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)})) = ∩ {𝑥 ∣ ((∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
51	42, 50	eqtrd 2777	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (t+‘((t+‘𝑅) ∪ (t+‘𝑆))) = ∩ {𝑥 ∣ ((∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
52	31, 34, 51	3eqtr4a 2803	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (t+‘(𝑅 ∪ 𝑆)) = (t+‘((t+‘𝑅) ∪ (t+‘𝑆))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2714  Vcvv 3480   ∪ cun 3949   ⊆ wss 3951  ∩ cint 4946   ∘ ccom 5689  ‘cfv 6561  t+ctcl 15024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-trcl 15026

This theorem is referenced by: (None)