Theorem trclun 15077

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 4146	. . . . . . . . . 10 ⊢ ((𝑅 ⊆ 𝑥 ∧ 𝑆 ⊆ 𝑥) ↔ (𝑅 ∪ 𝑆) ⊆ 𝑥)
2		simpl 488	. . . . . . . . . 10 ⊢ ((𝑅 ⊆ 𝑥 ∧ 𝑆 ⊆ 𝑥) → 𝑅 ⊆ 𝑥)
3	1, 2	sylbir 238	. . . . . . . . 9 ⊢ ((𝑅 ∪ 𝑆) ⊆ 𝑥 → 𝑅 ⊆ 𝑥)
4		vex 3462	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
5		trcleq2lem 15054	. . . . . . . . . . 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 592	. . . . . . . 8 ⊢ (((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) → 𝑥 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)})
9		intss1 4933	. . . . . . . 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 490	. . . . . . . . . 10 ⊢ ((𝑅 ⊆ 𝑥 ∧ 𝑆 ⊆ 𝑥) → 𝑆 ⊆ 𝑥)
12	1, 11	sylbir 238	. . . . . . . . 9 ⊢ ((𝑅 ∪ 𝑆) ⊆ 𝑥 → 𝑆 ⊆ 𝑥)
13		trcleq2lem 15054	. . . . . . . . . . 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 592	. . . . . . . 8 ⊢ (((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) → 𝑥 ∈ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)})
17		intss1 4933	. . . . . . . 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 4148	. . . . . 6 ⊢ (((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) → (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}) ⊆ 𝑥)
20		simpr 490	. . . . . 6 ⊢ (((𝑅 ∪ 𝑆) ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) → (𝑥 ∘ 𝑥) ⊆ 𝑥)
21	19, 20	jca 521	. . . . 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 4937	. . . . . . . 8 ⊢ 𝑅 ⊆ ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}
23		ssmin 4937	. . . . . . . 8 ⊢ 𝑆 ⊆ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}
24		unss12 4144	. . . . . . . 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 705	. . . . . . 7 ⊢ (𝑅 ∪ 𝑆) ⊆ (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)})
26		sstr 3948	. . . . . . 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 703	. . . . . 6 ⊢ ((∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}) ⊆ 𝑥 → (𝑅 ∪ 𝑆) ⊆ 𝑥)
28	27	anim1i 627	. . . . 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 2833	. . 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 4921	. 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 7754	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ∪ 𝑆) ∈ V)
33		trclfv 15063	. . 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 488	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑅 ∈ 𝑉)
36		trclfv 15063	. . . . . 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 490	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑆 ∈ 𝑊)
39		trclfv 15063	. . . . . 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 4126	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((t+‘𝑅) ∪ (t+‘𝑆)) = (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}))
42	41	fveq2d 6892	. . 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 6901	. . . . . 6 ⊢ (t+‘𝑅) ∈ V
44	36, 43	eqeltrrdi 2875	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∈ V)
45		fvex 6901	. . . . . 6 ⊢ (t+‘𝑆) ∈ V
46	39, 45	eqeltrrdi 2875	. . . . 5 ⊢ (𝑆 ∈ 𝑊 → ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)} ∈ V)
47		unexg 7754	. . . . 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 608	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ∪ ∩ {𝑠 ∣ (𝑆 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}) ∈ V)
49		trclfv 15063	. . . 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 2801	. 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 2827	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (t+‘(𝑅 ∪ 𝑆)) = (t+‘((t+‘𝑅) ∪ (t+‘𝑆))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {cab 2744  Vcvv 3458   ∪ cun 3906   ⊆ wss 3908  ∩ cint 4917   ∘ ccom 5670  ‘cfv 6543  t+ctcl 15048

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-iota 6499  df-fun 6545  df-fv 6551  df-trcl 15050

This theorem is used by: (None)