Theorem dfrtrcl2 14421

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 2822	. . . 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 5772	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑅 → dom 𝑥 = dom 𝑅)
3		rneq 5806	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑅 → ran 𝑥 = ran 𝑅)
4	2, 3	uneq12d 4140	. . . . . . . . . 10 ⊢ (𝑥 = 𝑅 → (dom 𝑥 ∪ ran 𝑥) = (dom 𝑅 ∪ ran 𝑅))
5	4	reseq2d 5853	. . . . . . . . 9 ⊢ (𝑥 = 𝑅 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
6	5	sseq1d 3998	. . . . . . . 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 3998	. . . . . . . 8 ⊢ (𝑥 = 𝑅 → (𝑥 ⊆ 𝑧 ↔ 𝑅 ⊆ 𝑧))
9	6, 8	3anbi12d 1433	. . . . . . 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 2885	. . . . . 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 4881	. . . . 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 484	. . . 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 6126	. . . . . . . . . 10 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
16	14, 15	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
17	16	eqcomd 2827	. . . . . . . 8 ⊢ (𝜑 → (dom 𝑅 ∪ ran 𝑅) = ∪ ∪ 𝑅)
18	14, 13	rtrclreclem1 14417	. . . . . . . . 9 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅))
19		id 22	. . . . . . . . . . 11 ⊢ ((dom 𝑅 ∪ ran 𝑅) = ∪ ∪ 𝑅 → (dom 𝑅 ∪ ran 𝑅) = ∪ ∪ 𝑅)
20	19	reseq2d 5853	. . . . . . . . . 10 ⊢ ((dom 𝑅 ∪ ran 𝑅) = ∪ ∪ 𝑅 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ ∪ ∪ 𝑅))
21	20	sseq1d 3998	. . . . . . . . 9 ⊢ ((dom 𝑅 ∪ ran 𝑅) = ∪ ∪ 𝑅 → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ↔ ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅)))
22	18, 21	syl5ibr 248	. . . . . . . 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 14418	. . . . . . 7 ⊢ (𝜑 → 𝑅 ⊆ (t*rec‘𝑅))
25	14, 13	rtrclreclem3 14419	. . . . . . 7 ⊢ (𝜑 → ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))
26		fvex 6683	. . . . . . . 8 ⊢ (t*rec‘𝑅) ∈ V
27		sseq2 3993	. . . . . . . . . . 11 ⊢ (𝑧 = (t*rec‘𝑅) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅)))
28		sseq2 3993	. . . . . . . . . . 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 5735	. . . . . . . . . . . 12 ⊢ (𝑧 = (t*rec‘𝑅) → (𝑧 ∘ 𝑧) = ((t*rec‘𝑅) ∘ (t*rec‘𝑅)))
31	30, 29	sseq12d 4000	. . . . . . . . . . 11 ⊢ (𝑧 = (t*rec‘𝑅) → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅)))
32	27, 28, 31	3anbi123d 1432	. . . . . . . . . 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 1928	. . . . . . . 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 3663	. . . . . . . 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 589	. . . . . . 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 1338	. . . . . 6 ⊢ (𝜑 → (t*rec‘𝑅) ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
38	37	ne0d 4301	. . . . 5 ⊢ (𝜑 → {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ≠ ∅)
39		intex 5240	. . . . 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 220	. . . 4 ⊢ (𝜑 → ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ∈ V)
41	1, 12, 13, 40	fvmptd 6775	. . 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 4891	. . . . 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 3497	. . . . . . . 8 ⊢ 𝑠 ∈ V
45		sseq2 3993	. . . . . . . . 9 ⊢ (𝑧 = 𝑠 → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))
46		sseq2 3993	. . . . . . . . 9 ⊢ (𝑧 = 𝑠 → (𝑅 ⊆ 𝑧 ↔ 𝑅 ⊆ 𝑠))
47		id 22	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑠 → 𝑧 = 𝑠)
48	47, 47	coeq12d 5735	. . . . . . . . . 10 ⊢ (𝑧 = 𝑠 → (𝑧 ∘ 𝑧) = (𝑠 ∘ 𝑠))
49	48, 47	sseq12d 4000	. . . . . . . . 9 ⊢ (𝑧 = 𝑠 → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ (𝑠 ∘ 𝑠) ⊆ 𝑠))
50	45, 46, 49	3anbi123d 1432	. . . . . . . 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 3667	. . . . . . 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 14420	. . . . . . . 8 ⊢ (𝜑 → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
53	52	19.21bi 2188	. . . . . . 7 ⊢ (𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
54	51, 53	syl5bi 244	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} → (t*rec‘𝑅) ⊆ 𝑠))
55	54	ralrimiv 3181	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} (t*rec‘𝑅) ⊆ 𝑠)
56		ssint 4892	. . . . 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 236	. . . 4 ⊢ (𝜑 → (t*rec‘𝑅) ⊆ ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
58	43, 57	eqssd 3984	. . 3 ⊢ (𝜑 → ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = (t*rec‘𝑅))
59	41, 58	eqtrd 2856	. 2 ⊢ (𝜑 → ((𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})‘𝑅) = (t*rec‘𝑅))
60		df-rtrcl 14348	. . 3 ⊢ t* = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
61		fveq1 6669	. . . . 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 2823	. . . 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 343	. . 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 233	1 ⊢ (𝜑 → (t*‘𝑅) = (t*rec‘𝑅))

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1083  ∀wal 1535   = wceq 1537   ∈ wcel 2114  {cab 2799   ≠ wne 3016  ∀wral 3138  Vcvv 3494   ∪ cun 3934   ⊆ wss 3936  ∅c0 4291  ∪ cuni 4838  ∩ cint 4876   ↦ cmpt 5146   I cid 5459  dom cdm 5555  ran crn 5556   ↾ cres 5557   ∘ ccom 5559  Rel wrel 5560  ‘cfv 6355  t*crtcl 14346  t*reccrtrcl 14414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-n0 11899  df-z 11983  df-uz 12245  df-seq 13371  df-rtrcl 14348  df-relexp 14380  df-rtrclrec 14415

This theorem is referenced by:  rtrclind  14424