Theorem trclun 15041

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 4145	. . . . . . . . . 10 ⊢ ((𝑅 ⊆ 𝑥 ∧ 𝑆 ⊆ 𝑥) ↔ (𝑅 ∪ 𝑆) ⊆ 𝑥)
2		simpl 487	. . . . . . . . . 10 ⊢ ((𝑅 ⊆ 𝑥 ∧ 𝑆 ⊆ 𝑥) → 𝑅 ⊆ 𝑥)
3	1, 2	sylbir 238	. . . . . . . . 9 ⊢ ((𝑅 ∪ 𝑆) ⊆ 𝑥 → 𝑅 ⊆ 𝑥)
4		vex 3461	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
5		trcleq2lem 15018	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑥 → ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ↔ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)))
6	4, 5	elab 3641	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ↔ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥))
7	6	biimpri 231	. . . . . . . . 9 ⊢ ((𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) → 𝑥 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)})
8	3, 7	sylan 591	. . . . . . . 8 ⊢ (((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) → 𝑥 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)})
9		intss1 4924	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ⊆ 𝑥)
10	8, 9	syl 18	. . . . . . 7 ⊢ (((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ⊆ 𝑥)
11		simpr 489	. . . . . . . . . 10 ⊢ ((𝑅 ⊆ 𝑥 ∧ 𝑆 ⊆ 𝑥) → 𝑆 ⊆ 𝑥)
12	1, 11	sylbir 238	. . . . . . . . 9 ⊢ ((𝑅 ∪ 𝑆) ⊆ 𝑥 → 𝑆 ⊆ 𝑥)
13		trcleq2lem 15018	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑥 → ((𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) ↔ (𝑆 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)))
14	4, 13	elab 3641	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)} ↔ (𝑆 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥))
15	14	biimpri 231	. . . . . . . . 9 ⊢ ((𝑆 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) → 𝑥 ∈ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)})
16	12, 15	sylan 591	. . . . . . . 8 ⊢ (((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) → 𝑥 ∈ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)})
17		intss1 4924	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)} → ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)} ⊆ 𝑥)
18	16, 17	syl 18	. . . . . . 7 ⊢ (((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) → ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)} ⊆ 𝑥)
19	10, 18	unssd 4147	. . . . . 6 ⊢ (((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) → (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}) ⊆ 𝑥)
20		simpr 489	. . . . . 6 ⊢ (((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) → (𝑥 ∘ 𝑥) ⊆ 𝑥)
21	19, 20	jca 520	. . . . 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 4928	. . . . . . . 8 ⊢ 𝑅 ⊆ ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}
23		ssmin 4928	. . . . . . . 8 ⊢ 𝑆 ⊆ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}
24		unss12 4143	. . . . . . . 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 704	. . . . . . 7 ⊢ (𝑅 ∪ 𝑆) ⊆ (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)})
26		sstr 3947	. . . . . . 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 702	. . . . . 6 ⊢ ((∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}) ⊆ 𝑥 → (𝑅 ∪ 𝑆) ⊆ 𝑥)
28	27	anim1i 626	. . . . 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 212	. . . 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 2832	. . 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 4912	. 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 7730	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ∪ 𝑆) ∈ V)
33		trclfv 15027	. . 3 ⊢ ((𝑅 ∪ 𝑆) ∈ V → (t+‘(𝑅 ∪ 𝑆)) = ∩ {𝑥 ∣ ((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
34	32, 33	syl 18	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (t+‘(𝑅 ∪ 𝑆)) = ∩ {𝑥 ∣ ((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
35		simpl 487	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑅 ∈ 𝑉)
36		trclfv 15027	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)})
37	35, 36	syl 18	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (t+‘𝑅) = ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)})
38		simpr 489	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑆 ∈ 𝑊)
39		trclfv 15027	. . . . . 6 ⊢ (𝑆 ∈ 𝑊 → (t+‘𝑆) = ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)})
40	38, 39	syl 18	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (t+‘𝑆) = ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)})
41	37, 40	uneq12d 4125	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((t+‘𝑅) ∪ (t+‘𝑆)) = (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}))
42	41	fveq2d 6875	. . 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 6884	. . . . . 6 ⊢ (t+‘𝑅) ∈ V
44	36, 43	eqeltrrdi 2874	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∈ V)
45		fvex 6884	. . . . . 6 ⊢ (t+‘𝑆) ∈ V
46	39, 45	eqeltrrdi 2874	. . . . 5 ⊢ (𝑆 ∈ 𝑊 → ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)} ∈ V)
47		unexg 7730	. . . . 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 607	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}) ∈ V)
49		trclfv 15027	. . . 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 18	. . 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 2800	. 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 2826	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (t+‘(𝑅 ∪ 𝑆)) = (t+‘((t+‘𝑅) ∪ (t+‘𝑆))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  {cab 2743  Vcvv 3457   ∪ cun 3905   ⊆ wss 3907  ∩ cint 4908   ∘ ccom 5656  ‘cfv 6525  t+ctcl 15012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-iota 6481  df-fun 6527  df-fv 6533  df-trcl 15014

This theorem is referenced by: (None)