Theorem dfrtrcl2 14209

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.)

Hypotheses

Ref	Expression
drrtrcl2.1	⊢ (𝜑 → Rel 𝑅)
drrtrcl2.2	⊢ (𝜑 → 𝑅 ∈ V)

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 2779	. . . 4 ⊢ (𝜑 → (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 5569	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑅 → dom 𝑥 = dom 𝑅)
3		rneq 5596	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑅 → ran 𝑥 = ran 𝑅)
4	2, 3	uneq12d 3991	. . . . . . . . . 10 ⊢ (𝑥 = 𝑅 → (dom 𝑥 ∪ ran 𝑥) = (dom 𝑅 ∪ ran 𝑅))
5	4	reseq2d 5642	. . . . . . . . 9 ⊢ (𝑥 = 𝑅 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
6	5	sseq1d 3851	. . . . . . . 8 ⊢ (𝑥 = 𝑅 → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧))
7		id 22	. . . . . . . . 9 ⊢ (𝑥 = 𝑅 → 𝑥 = 𝑅)
8	7	sseq1d 3851	. . . . . . . 8 ⊢ (𝑥 = 𝑅 → (𝑥 ⊆ 𝑧 ↔ 𝑅 ⊆ 𝑧))
9	6, 8	3anbi12d 1510	. . . . . . 7 ⊢ (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 2906	. . . . . 6 ⊢ (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 4715	. . . . 5 ⊢ (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 475	. . . 4 ⊢ ((𝜑 ∧ 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		drrtrcl2.2	. . . 4 ⊢ (𝜑 → 𝑅 ∈ V)
14		drrtrcl2.1	. . . . . . . . . 10 ⊢ (𝜑 → Rel 𝑅)
15		relfld 5915	. . . . . . . . . 10 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
16	14, 15	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
17	16	eqcomd 2784	. . . . . . . 8 ⊢ (𝜑 → (dom 𝑅 ∪ ran 𝑅) = ∪ ∪ 𝑅)
18	14, 13	rtrclreclem1 14205	. . . . . . . . 9 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅))
19		id 22	. . . . . . . . . . 11 ⊢ ((dom 𝑅 ∪ ran 𝑅) = ∪ ∪ 𝑅 → (dom 𝑅 ∪ ran 𝑅) = ∪ ∪ 𝑅)
20	19	reseq2d 5642	. . . . . . . . . 10 ⊢ ((dom 𝑅 ∪ ran 𝑅) = ∪ ∪ 𝑅 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ ∪ ∪ 𝑅))
21	20	sseq1d 3851	. . . . . . . . 9 ⊢ ((dom 𝑅 ∪ ran 𝑅) = ∪ ∪ 𝑅 → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ↔ ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅)))
22	18, 21	syl5ibr 238	. . . . . . . 8 ⊢ ((dom 𝑅 ∪ ran 𝑅) = ∪ ∪ 𝑅 → (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅)))
23	17, 22	mpcom 38	. . . . . . 7 ⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅))
24	13	rtrclreclem2 14206	. . . . . . 7 ⊢ (𝜑 → 𝑅 ⊆ (t*rec‘𝑅))
25	14, 13	rtrclreclem3 14207	. . . . . . 7 ⊢ (𝜑 → ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))
26		fvex 6459	. . . . . . . 8 ⊢ (t*rec‘𝑅) ∈ V
27		sseq2 3846	. . . . . . . . . . 11 ⊢ (𝑧 = (t*rec‘𝑅) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅)))
28		sseq2 3846	. . . . . . . . . . 11 ⊢ (𝑧 = (t*rec‘𝑅) → (𝑅 ⊆ 𝑧 ↔ 𝑅 ⊆ (t*rec‘𝑅)))
29		id 22	. . . . . . . . . . . . 13 ⊢ (𝑧 = (t*rec‘𝑅) → 𝑧 = (t*rec‘𝑅))
30	29, 29	coeq12d 5532	. . . . . . . . . . . 12 ⊢ (𝑧 = (t*rec‘𝑅) → (𝑧 ∘ 𝑧) = ((t*rec‘𝑅) ∘ (t*rec‘𝑅)))
31	30, 29	sseq12d 3853	. . . . . . . . . . 11 ⊢ (𝑧 = (t*rec‘𝑅) → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅)))
32	27, 28, 31	3anbi123d 1509	. . . . . . . . . 10 ⊢ (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))))
33	32	a1i 11	. . . . . . . . 9 ⊢ (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)))))
34	33	alrimiv 1970	. . . . . . . 8 ⊢ (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)))))
35		elabgt 3553	. . . . . . . 8 ⊢ (((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))))
36	26, 34, 35	sylancr 581	. . . . . . 7 ⊢ (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))))
37	23, 24, 25, 36	mpbir3and 1399	. . . . . 6 ⊢ (𝜑 → (t*rec‘𝑅) ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
38	37	ne0d 4150	. . . . 5 ⊢ (𝜑 → {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ≠ ∅)
39		intex 5054	. . . . 5 ⊢ ({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)
40	38, 39	sylib 210	. . . 4 ⊢ (𝜑 → ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ∈ V)
41	1, 12, 13, 40	fvmptd 6548	. . 3 ⊢ (𝜑 → ((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)})
42		intss1 4725	. . . . 5 ⊢ ((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))
43	37, 42	syl 17	. . . 4 ⊢ (𝜑 → ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ⊆ (t*rec‘𝑅))
44		vex 3401	. . . . . . . 8 ⊢ 𝑠 ∈ V
45		sseq2 3846	. . . . . . . . 9 ⊢ (𝑧 = 𝑠 → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))
46		sseq2 3846	. . . . . . . . 9 ⊢ (𝑧 = 𝑠 → (𝑅 ⊆ 𝑧 ↔ 𝑅 ⊆ 𝑠))
47		id 22	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑠 → 𝑧 = 𝑠)
48	47, 47	coeq12d 5532	. . . . . . . . . 10 ⊢ (𝑧 = 𝑠 → (𝑧 ∘ 𝑧) = (𝑠 ∘ 𝑠))
49	48, 47	sseq12d 3853	. . . . . . . . 9 ⊢ (𝑧 = 𝑠 → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ (𝑠 ∘ 𝑠) ⊆ 𝑠))
50	45, 46, 49	3anbi123d 1509	. . . . . . . 8 ⊢ (𝑧 = 𝑠 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)))
51	44, 50	elab 3558	. . . . . . 7 ⊢ (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠))
52	14, 13	rtrclreclem4 14208	. . . . . . . 8 ⊢ (𝜑 → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
53	52	19.21bi 2173	. . . . . . 7 ⊢ (𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
54	51, 53	syl5bi 234	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} → (t*rec‘𝑅) ⊆ 𝑠))
55	54	ralrimiv 3147	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} (t*rec‘𝑅) ⊆ 𝑠)
56		ssint 4726	. . . . 5 ⊢ ((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)
57	55, 56	sylibr 226	. . . 4 ⊢ (𝜑 → (t*rec‘𝑅) ⊆ ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
58	43, 57	eqssd 3838	. . 3 ⊢ (𝜑 → ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = (t*rec‘𝑅))
59	41, 58	eqtrd 2814	. 2 ⊢ (𝜑 → ((𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})‘𝑅) = (t*rec‘𝑅))
60		df-rtrcl 14136	. . 3 ⊢ t* = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
61		fveq1 6445	. . . . 5 ⊢ (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))
62	61	eqeq1d 2780	. . . 4 ⊢ (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)))
63	62	imbi2d 332	. . 3 ⊢ (t* = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) → ((𝜑 → (t*‘𝑅) = (t*rec‘𝑅)) ↔ (𝜑 → ((𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})‘𝑅) = (t*rec‘𝑅))))
64	60, 63	ax-mp 5	. 2 ⊢ ((𝜑 → (t*‘𝑅) = (t*rec‘𝑅)) ↔ (𝜑 → ((𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})‘𝑅) = (t*rec‘𝑅)))
65	59, 64	mpbir 223	1 ⊢ (𝜑 → (t*‘𝑅) = (t*rec‘𝑅))

Syntax hints:   → wi 4   ↔ wb 198   ∧ w3a 1071  ∀wal 1599   = wceq 1601   ∈ wcel 2107  {cab 2763   ≠ wne 2969  ∀wral 3090  Vcvv 3398   ∪ cun 3790   ⊆ wss 3792  ∅c0 4141  ∪ cuni 4671  ∩ cint 4710   ↦ cmpt 4965   I cid 5260  dom cdm 5355  ran crn 5356   ↾ cres 5357   ∘ ccom 5359  Rel wrel 5360  ‘cfv 6135  t*crtcl 14134  t*reccrtrcl 14202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-n0 11643  df-z 11729  df-uz 11993  df-seq 13120  df-rtrcl 14136  df-relexp 14168  df-rtrclrec 14203

This theorem is referenced by:  rtrclind  14212