Theorem dfrtrcl2 14773

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 2739	. . . . . 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 5812	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑅 → dom 𝑥 = dom 𝑅)
3		rneq 5845	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑅 → ran 𝑥 = ran 𝑅)
4	2, 3	uneq12d 4098	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑅 → (dom 𝑥 ∪ ran 𝑥) = (dom 𝑅 ∪ ran 𝑅))
5	4	reseq2d 5891	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑅 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
6	5	sseq1d 3952	. . . . . . . . . 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 3952	. . . . . . . . . 10 ⊢ (𝑥 = 𝑅 → (𝑥 ⊆ 𝑧 ↔ 𝑅 ⊆ 𝑧))
9	6, 8	3anbi12d 1436	. . . . . . . . 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 4884	. . . . . . 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 482	. . . . . 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 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → 𝑅 ∈ V)
14		dfrtrcl2.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → Rel 𝑅)
15		relfld 6178	. . . . . . . . . . . . 13 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
16	14, 15	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
17	16	eqcomd 2744	. . . . . . . . . . 11 ⊢ (𝜑 → (dom 𝑅 ∪ ran 𝑅) = ∪ ∪ 𝑅)
18	17	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) = ∪ ∪ 𝑅)
19	14	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → Rel 𝑅)
20	19, 13	rtrclreclem2 14770	. . . . . . . . . . 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 5891	. . . . . . . . . . . 12 ⊢ ((dom 𝑅 ∪ ran 𝑅) = ∪ ∪ 𝑅 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ ∪ ∪ 𝑅))
23	22	sseq1d 3952	. . . . . . . . . . 11 ⊢ ((dom 𝑅 ∪ ran 𝑅) = ∪ ∪ 𝑅 → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ↔ ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅)))
24	20, 23	syl5ibr 245	. . . . . . . . . 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 14768	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → 𝑅 ⊆ (t*rec‘𝑅))
27	14	rtrclreclem3 14771	. . . . . . . . . 10 ⊢ (𝜑 → ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))
28	27	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))
29		fvex 6787	. . . . . . . . . . 11 ⊢ (t*rec‘𝑅) ∈ V
30		sseq2 3947	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (t*rec‘𝑅) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅)))
31		sseq2 3947	. . . . . . . . . . . . . 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 5773	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (t*rec‘𝑅) → (𝑧 ∘ 𝑧) = ((t*rec‘𝑅) ∘ (t*rec‘𝑅)))
34	33, 32	sseq12d 3954	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (t*rec‘𝑅) → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅)))
35	30, 31, 34	3anbi123d 1435	. . . . . . . . . . . . 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 1930	. . . . . . . . . . 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 3603	. . . . . . . . . . 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 481	. . . . . . . . 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 1341	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → (t*rec‘𝑅) ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
42	41	ne0d 4269	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ≠ ∅)
43		intex 5261	. . . . . . 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 217	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ∈ V)
45	1, 12, 13, 44	fvmptd 6882	. . . . 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 4894	. . . . . . 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 3436	. . . . . . . . . . 11 ⊢ 𝑠 ∈ V
49		sseq2 3947	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑠 → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))
50		sseq2 3947	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑠 → (𝑅 ⊆ 𝑧 ↔ 𝑅 ⊆ 𝑠))
51		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑠 → 𝑧 = 𝑠)
52	51, 51	coeq12d 5773	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑠 → (𝑧 ∘ 𝑧) = (𝑠 ∘ 𝑠))
53	52, 51	sseq12d 3954	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑠 → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ (𝑠 ∘ 𝑠) ⊆ 𝑠))
54	49, 50, 53	3anbi123d 1435	. . . . . . . . . . 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 3609	. . . . . . . . . 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 14772	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
57	56	19.21bi 2182	. . . . . . . . . 10 ⊢ (𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
58	55, 57	syl5bi 241	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} → (t*rec‘𝑅) ⊆ 𝑠))
59	58	ralrimiv 3102	. . . . . . . 8 ⊢ (𝜑 → ∀𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} (t*rec‘𝑅) ⊆ 𝑠)
60		ssint 4895	. . . . . . . 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 233	. . . . . . 7 ⊢ (𝜑 → (t*rec‘𝑅) ⊆ ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
62	61	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → (t*rec‘𝑅) ⊆ ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
63	47, 62	eqssd 3938	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = (t*rec‘𝑅))
64	45, 63	eqtrd 2778	. . . 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 14699	. . . . 5 ⊢ t* = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
66		fveq1 6773	. . . . . . 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 2740	. . . . . 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 341	. . . . 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 230	. . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → (t*‘𝑅) = (t*rec‘𝑅))
71	70	ex 413	. 2 ⊢ (𝜑 → (𝑅 ∈ V → (t*‘𝑅) = (t*rec‘𝑅)))
72		fvprc 6766	. . 3 ⊢ (¬ 𝑅 ∈ V → (t*‘𝑅) = ∅)
73		fvprc 6766	. . . 4 ⊢ (¬ 𝑅 ∈ V → (t*rec‘𝑅) = ∅)
74	73	eqcomd 2744	. . 3 ⊢ (¬ 𝑅 ∈ V → ∅ = (t*rec‘𝑅))
75	72, 74	eqtrd 2778	. 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 205   ∧ wa 396   ∧ w3a 1086  ∀wal 1537   = wceq 1539   ∈ wcel 2106  {cab 2715   ≠ wne 2943  ∀wral 3064  Vcvv 3432   ∪ cun 3885   ⊆ wss 3887  ∅c0 4256  ∪ cuni 4839  ∩ cint 4879   ↦ cmpt 5157   I cid 5488  dom cdm 5589  ran crn 5590   ↾ cres 5591   ∘ ccom 5593  Rel wrel 5594  ‘cfv 6433  t*crtcl 14697  t*reccrtrcl 14766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-seq 13722  df-rtrcl 14699  df-relexp 14731  df-rtrclrec 14767

This theorem is referenced by:  rtrclind  14776