Theorem dfrtrcl2 15102

Description: The two definitions t* and t*rec of the reflexive, transitive closure coincide if 𝑅 is indeed a relation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 13-Jul-2024.)

Hypothesis

Ref	Expression
dfrtrcl2.1	⊢ (𝜑 → Rel 𝑅)

Assertion

Ref	Expression
dfrtrcl2	⊢ (𝜑 → (t*‘𝑅) = (t*rec‘𝑅))

Proof of Theorem dfrtrcl2

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

Step	Hyp	Ref	Expression
1		eqidd 2737	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}))
2		dmeq 5913	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑅 → dom 𝑥 = dom 𝑅)
3		rneq 5946	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑅 → ran 𝑥 = ran 𝑅)
4	2, 3	uneq12d 4168	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑅 → (dom 𝑥 ∪ ran 𝑥) = (dom 𝑅 ∪ ran 𝑅))
5	4	reseq2d 5996	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑅 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
6	5	sseq1d 4014	. . . . . . . . . 10 ⊢ (𝑥 = 𝑅 → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧))
7		id 22	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑅 → 𝑥 = 𝑅)
8	7	sseq1d 4014	. . . . . . . . . 10 ⊢ (𝑥 = 𝑅 → (𝑥 ⊆ 𝑧 ↔ 𝑅 ⊆ 𝑧))
9	6, 8	3anbi12d 1438	. . . . . . . . 9 ⊢ (𝑥 = 𝑅 → ((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)))
10	9	abbidv 2807	. . . . . . . 8 ⊢ (𝑥 = 𝑅 → {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
11	10	inteqd 4950	. . . . . . 7 ⊢ (𝑥 = 𝑅 → ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
12	11	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ∈ V) ∧ 𝑥 = 𝑅) → ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
13		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → 𝑅 ∈ V)
14		dfrtrcl2.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → Rel 𝑅)
15		relfld 6294	. . . . . . . . . . . . 13 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
16	14, 15	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
17	16	eqcomd 2742	. . . . . . . . . . 11 ⊢ (𝜑 → (dom 𝑅 ∪ ran 𝑅) = ∪ ∪ 𝑅)
18	17	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) = ∪ ∪ 𝑅)
19	14	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → Rel 𝑅)
20	19, 13	rtrclreclem2 15099	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅))
21		id 22	. . . . . . . . . . . . 13 ⊢ ((dom 𝑅 ∪ ran 𝑅) = ∪ ∪ 𝑅 → (dom 𝑅 ∪ ran 𝑅) = ∪ ∪ 𝑅)
22	21	reseq2d 5996	. . . . . . . . . . . 12 ⊢ ((dom 𝑅 ∪ ran 𝑅) = ∪ ∪ 𝑅 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ ∪ ∪ 𝑅))
23	22	sseq1d 4014	. . . . . . . . . . 11 ⊢ ((dom 𝑅 ∪ ran 𝑅) = ∪ ∪ 𝑅 → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ↔ ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅)))
24	20, 23	imbitrrid 246	. . . . . . . . . 10 ⊢ ((dom 𝑅 ∪ ran 𝑅) = ∪ ∪ 𝑅 → ((𝜑 ∧ 𝑅 ∈ V) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅)))
25	18, 24	mpcom 38	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅))
26	13	rtrclreclem1 15097	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → 𝑅 ⊆ (t*rec‘𝑅))
27	14	rtrclreclem3 15100	. . . . . . . . . 10 ⊢ (𝜑 → ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))
28	27	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))
29		fvex 6918	. . . . . . . . . . 11 ⊢ (t*rec‘𝑅) ∈ V
30		sseq2 4009	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (t*rec‘𝑅) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅)))
31		sseq2 4009	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (t*rec‘𝑅) → (𝑅 ⊆ 𝑧 ↔ 𝑅 ⊆ (t*rec‘𝑅)))
32		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (t*rec‘𝑅) → 𝑧 = (t*rec‘𝑅))
33	32, 32	coeq12d 5874	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (t*rec‘𝑅) → (𝑧 ∘ 𝑧) = ((t*rec‘𝑅) ∘ (t*rec‘𝑅)))
34	33, 32	sseq12d 4016	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (t*rec‘𝑅) → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅)))
35	30, 31, 34	3anbi123d 1437	. . . . . . . . . . . . 13 ⊢ (𝑧 = (t*rec‘𝑅) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ∧ 𝑅 ⊆ (t*rec‘𝑅) ∧ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))))
36	35	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 = (t*rec‘𝑅) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ∧ 𝑅 ⊆ (t*rec‘𝑅) ∧ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅)))))
37	36	alrimiv 1926	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑧(𝑧 = (t*rec‘𝑅) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ∧ 𝑅 ⊆ (t*rec‘𝑅) ∧ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅)))))
38		elabgt 3671	. . . . . . . . . . 11 ⊢ (((t*rec‘𝑅) ∈ V ∧ ∀𝑧(𝑧 = (t*rec‘𝑅) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ∧ 𝑅 ⊆ (t*rec‘𝑅) ∧ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))))) → ((t*rec‘𝑅) ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ∧ 𝑅 ⊆ (t*rec‘𝑅) ∧ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))))
39	29, 37, 38	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → ((t*rec‘𝑅) ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ∧ 𝑅 ⊆ (t*rec‘𝑅) ∧ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))))
40	39	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → ((t*rec‘𝑅) ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ∧ 𝑅 ⊆ (t*rec‘𝑅) ∧ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))))
41	25, 26, 28, 40	mpbir3and 1342	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → (t*rec‘𝑅) ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
42	41	ne0d 4341	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ≠ ∅)
43		intex 5343	. . . . . . 7 ⊢ ({𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ≠ ∅ ↔ ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ∈ V)
44	42, 43	sylib 218	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ∈ V)
45	1, 12, 13, 44	fvmptd 7022	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → ((𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})‘𝑅) = ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
46		intss1 4962	. . . . . . 7 ⊢ ((t*rec‘𝑅) ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} → ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ⊆ (t*rec‘𝑅))
47	41, 46	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ⊆ (t*rec‘𝑅))
48		vex 3483	. . . . . . . . . . 11 ⊢ 𝑠 ∈ V
49		sseq2 4009	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑠 → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))
50		sseq2 4009	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑠 → (𝑅 ⊆ 𝑧 ↔ 𝑅 ⊆ 𝑠))
51		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑠 → 𝑧 = 𝑠)
52	51, 51	coeq12d 5874	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑠 → (𝑧 ∘ 𝑧) = (𝑠 ∘ 𝑠))
53	52, 51	sseq12d 4016	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑠 → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ (𝑠 ∘ 𝑠) ⊆ 𝑠))
54	49, 50, 53	3anbi123d 1437	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑠 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)))
55	48, 54	elab 3678	. . . . . . . . . 10 ⊢ (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠))
56	14	rtrclreclem4 15101	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
57	56	19.21bi 2188	. . . . . . . . . 10 ⊢ (𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
58	55, 57	biimtrid 242	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} → (t*rec‘𝑅) ⊆ 𝑠))
59	58	ralrimiv 3144	. . . . . . . 8 ⊢ (𝜑 → ∀𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} (t*rec‘𝑅) ⊆ 𝑠)
60		ssint 4963	. . . . . . . 8 ⊢ ((t*rec‘𝑅) ⊆ ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ↔ ∀𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} (t*rec‘𝑅) ⊆ 𝑠)
61	59, 60	sylibr 234	. . . . . . 7 ⊢ (𝜑 → (t*rec‘𝑅) ⊆ ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
62	61	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → (t*rec‘𝑅) ⊆ ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
63	47, 62	eqssd 4000	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = (t*rec‘𝑅))
64	45, 63	eqtrd 2776	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → ((𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})‘𝑅) = (t*rec‘𝑅))
65		df-rtrcl 15028	. . . . 5 ⊢ t* = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
66		fveq1 6904	. . . . . . 7 ⊢ (t* = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) → (t*‘𝑅) = ((𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})‘𝑅))
67	66	eqeq1d 2738	. . . . . 6 ⊢ (t* = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) → ((t*‘𝑅) = (t*rec‘𝑅) ↔ ((𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})‘𝑅) = (t*rec‘𝑅)))
68	67	imbi2d 340	. . . . 5 ⊢ (t* = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) → (((𝜑 ∧ 𝑅 ∈ V) → (t*‘𝑅) = (t*rec‘𝑅)) ↔ ((𝜑 ∧ 𝑅 ∈ V) → ((𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})‘𝑅) = (t*rec‘𝑅))))
69	65, 68	ax-mp 5	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ∈ V) → (t*‘𝑅) = (t*rec‘𝑅)) ↔ ((𝜑 ∧ 𝑅 ∈ V) → ((𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})‘𝑅) = (t*rec‘𝑅)))
70	64, 69	mpbir 231	. . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → (t*‘𝑅) = (t*rec‘𝑅))
71	70	ex 412	. 2 ⊢ (𝜑 → (𝑅 ∈ V → (t*‘𝑅) = (t*rec‘𝑅)))
72		fvprc 6897	. . 3 ⊢ (¬ 𝑅 ∈ V → (t*‘𝑅) = ∅)
73		fvprc 6897	. . . 4 ⊢ (¬ 𝑅 ∈ V → (t*rec‘𝑅) = ∅)
74	73	eqcomd 2742	. . 3 ⊢ (¬ 𝑅 ∈ V → ∅ = (t*rec‘𝑅))
75	72, 74	eqtrd 2776	. 2 ⊢ (¬ 𝑅 ∈ V → (t*‘𝑅) = (t*rec‘𝑅))
76	71, 75	pm2.61d1 180	1 ⊢ (𝜑 → (t*‘𝑅) = (t*rec‘𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1537   = wceq 1539   ∈ wcel 2107  {cab 2713   ≠ wne 2939  ∀wral 3060  Vcvv 3479   ∪ cun 3948   ⊆ wss 3950  ∅c0 4332  ∪ cuni 4906  ∩ cint 4945   ↦ cmpt 5224   I cid 5576  dom cdm 5684  ran crn 5685   ↾ cres 5686   ∘ ccom 5688  Rel wrel 5689  ‘cfv 6560  t*crtcl 15026  t*reccrtrcl 15095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-n0 12529  df-z 12616  df-uz 12880  df-seq 14044  df-rtrcl 15028  df-relexp 15060  df-rtrclrec 15096

This theorem is referenced by:  rtrclind  15105