cnvrcl0 - Mathbox for Richard Penner

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Theorem cnvrcl0 43615

Description: The converse of the reflexive closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)

**Assertion**
Ref	Expression
cnvrcl0	⊢ (𝑋 ∈ 𝑉 → ^◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} = ∩ {𝑦 ∣ (^◡𝑋 ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)})

Distinct variable groups: 𝑥,𝑦,𝑉 𝑥,𝑋,𝑦

**Proof of Theorem cnvrcl0**
Step	Hyp	Ref	Expression
1		cnvresid 6647	. . . . . . 7 ⊢ ^◡( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom 𝑦 ∪ ran 𝑦))
2		cnvnonrel 43578	. . . . . . . . . . . . . . . 16 ⊢ ^◡(𝑋 ∖ ^◡^◡𝑋) = ∅
3		cnv0 6163	. . . . . . . . . . . . . . . 16 ⊢ ^◡∅ = ∅
4	2, 3	eqtr4i 2766	. . . . . . . . . . . . . . 15 ⊢ ^◡(𝑋 ∖ ^◡^◡𝑋) = ^◡∅
5	4	dmeqi 5918	. . . . . . . . . . . . . 14 ⊢ dom ^◡(𝑋 ∖ ^◡^◡𝑋) = dom ^◡∅
6		df-rn 5700	. . . . . . . . . . . . . 14 ⊢ ran (𝑋 ∖ ^◡^◡𝑋) = dom ^◡(𝑋 ∖ ^◡^◡𝑋)
7		df-rn 5700	. . . . . . . . . . . . . 14 ⊢ ran ∅ = dom ^◡∅
8	5, 6, 7	3eqtr4i 2773	. . . . . . . . . . . . 13 ⊢ ran (𝑋 ∖ ^◡^◡𝑋) = ran ∅
9		0ss 4406	. . . . . . . . . . . . . 14 ⊢ ∅ ⊆ ^◡𝑦
10	9	rnssi 5954	. . . . . . . . . . . . 13 ⊢ ran ∅ ⊆ ran ^◡𝑦
11	8, 10	eqsstri 4030	. . . . . . . . . . . 12 ⊢ ran (𝑋 ∖ ^◡^◡𝑋) ⊆ ran ^◡𝑦
12		ssequn2 4199	. . . . . . . . . . . 12 ⊢ (ran (𝑋 ∖ ^◡^◡𝑋) ⊆ ran ^◡𝑦 ↔ (ran ^◡𝑦 ∪ ran (𝑋 ∖ ^◡^◡𝑋)) = ran ^◡𝑦)
13	11, 12	mpbi 230	. . . . . . . . . . 11 ⊢ (ran ^◡𝑦 ∪ ran (𝑋 ∖ ^◡^◡𝑋)) = ran ^◡𝑦
14		rnun 6168	. . . . . . . . . . 11 ⊢ ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) = (ran ^◡𝑦 ∪ ran (𝑋 ∖ ^◡^◡𝑋))
15		dfdm4 5909	. . . . . . . . . . 11 ⊢ dom 𝑦 = ran ^◡𝑦
16	13, 14, 15	3eqtr4ri 2774	. . . . . . . . . 10 ⊢ dom 𝑦 = ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))
17	4	rneqi 5951	. . . . . . . . . . . . . 14 ⊢ ran ^◡(𝑋 ∖ ^◡^◡𝑋) = ran ^◡∅
18		dfdm4 5909	. . . . . . . . . . . . . 14 ⊢ dom (𝑋 ∖ ^◡^◡𝑋) = ran ^◡(𝑋 ∖ ^◡^◡𝑋)
19		dfdm4 5909	. . . . . . . . . . . . . 14 ⊢ dom ∅ = ran ^◡∅
20	17, 18, 19	3eqtr4i 2773	. . . . . . . . . . . . 13 ⊢ dom (𝑋 ∖ ^◡^◡𝑋) = dom ∅
21		dmss 5916	. . . . . . . . . . . . . 14 ⊢ (∅ ⊆ ^◡𝑦 → dom ∅ ⊆ dom ^◡𝑦)
22	9, 21	ax-mp 5	. . . . . . . . . . . . 13 ⊢ dom ∅ ⊆ dom ^◡𝑦
23	20, 22	eqsstri 4030	. . . . . . . . . . . 12 ⊢ dom (𝑋 ∖ ^◡^◡𝑋) ⊆ dom ^◡𝑦
24		ssequn2 4199	. . . . . . . . . . . 12 ⊢ (dom (𝑋 ∖ ^◡^◡𝑋) ⊆ dom ^◡𝑦 ↔ (dom ^◡𝑦 ∪ dom (𝑋 ∖ ^◡^◡𝑋)) = dom ^◡𝑦)
25	23, 24	mpbi 230	. . . . . . . . . . 11 ⊢ (dom ^◡𝑦 ∪ dom (𝑋 ∖ ^◡^◡𝑋)) = dom ^◡𝑦
26		dmun 5924	. . . . . . . . . . 11 ⊢ dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) = (dom ^◡𝑦 ∪ dom (𝑋 ∖ ^◡^◡𝑋))
27		df-rn 5700	. . . . . . . . . . 11 ⊢ ran 𝑦 = dom ^◡𝑦
28	25, 26, 27	3eqtr4ri 2774	. . . . . . . . . 10 ⊢ ran 𝑦 = dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))
29	16, 28	uneq12i 4176	. . . . . . . . 9 ⊢ (dom 𝑦 ∪ ran 𝑦) = (ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∪ dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))
30	29	equncomi 4170	. . . . . . . 8 ⊢ (dom 𝑦 ∪ ran 𝑦) = (dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∪ ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))
31	30	reseq2i 5997	. . . . . . 7 ⊢ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∪ ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))))
32	1, 31	eqtr2i 2764	. . . . . 6 ⊢ ( I ↾ (dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∪ ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))) = ^◡( I ↾ (dom 𝑦 ∪ ran 𝑦))
33		cnvss 5886	. . . . . 6 ⊢ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ^◡( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ ^◡𝑦)
34	32, 33	eqsstrid 4044	. . . . 5 ⊢ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∪ ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))) ⊆ ^◡𝑦)
35		ssun1 4188	. . . . 5 ⊢ ^◡𝑦 ⊆ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))
36	34, 35	sstrdi 4008	. . . 4 ⊢ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∪ ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))) ⊆ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))
37		dmeq 5917	. . . . . . 7 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → dom 𝑥 = dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))
38		rneq 5950	. . . . . . 7 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → ran 𝑥 = ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))
39	37, 38	uneq12d 4179	. . . . . 6 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → (dom 𝑥 ∪ ran 𝑥) = (dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∪ ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))))
40	39	reseq2d 6000	. . . . 5 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∪ ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))))
41		id 22	. . . . 5 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → 𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))
42	40, 41	sseq12d 4029	. . . 4 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 ↔ ( I ↾ (dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∪ ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))) ⊆ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))))
43	36, 42	imbitrrid 246	. . 3 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))
44	43	adantl 481	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))
45		cnvresid 6647	. . . . . 6 ⊢ ^◡( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑥 ∪ ran 𝑥))
46		dfdm4 5909	. . . . . . . . 9 ⊢ dom 𝑥 = ran ^◡𝑥
47		df-rn 5700	. . . . . . . . 9 ⊢ ran 𝑥 = dom ^◡𝑥
48	46, 47	uneq12i 4176	. . . . . . . 8 ⊢ (dom 𝑥 ∪ ran 𝑥) = (ran ^◡𝑥 ∪ dom ^◡𝑥)
49	48	equncomi 4170	. . . . . . 7 ⊢ (dom 𝑥 ∪ ran 𝑥) = (dom ^◡𝑥 ∪ ran ^◡𝑥)
50	49	reseq2i 5997	. . . . . 6 ⊢ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom ^◡𝑥 ∪ ran ^◡𝑥))
51	45, 50	eqtr2i 2764	. . . . 5 ⊢ ( I ↾ (dom ^◡𝑥 ∪ ran ^◡𝑥)) = ^◡( I ↾ (dom 𝑥 ∪ ran 𝑥))
52		cnvss 5886	. . . . 5 ⊢ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ^◡( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ ^◡𝑥)
53	51, 52	eqsstrid 4044	. . . 4 ⊢ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom ^◡𝑥 ∪ ran ^◡𝑥)) ⊆ ^◡𝑥)
54		dmeq 5917	. . . . . . 7 ⊢ (𝑦 = ^◡𝑥 → dom 𝑦 = dom ^◡𝑥)
55		rneq 5950	. . . . . . 7 ⊢ (𝑦 = ^◡𝑥 → ran 𝑦 = ran ^◡𝑥)
56	54, 55	uneq12d 4179	. . . . . 6 ⊢ (𝑦 = ^◡𝑥 → (dom 𝑦 ∪ ran 𝑦) = (dom ^◡𝑥 ∪ ran ^◡𝑥))
57	56	reseq2d 6000	. . . . 5 ⊢ (𝑦 = ^◡𝑥 → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom ^◡𝑥 ∪ ran ^◡𝑥)))
58		id 22	. . . . 5 ⊢ (𝑦 = ^◡𝑥 → 𝑦 = ^◡𝑥)
59	57, 58	sseq12d 4029	. . . 4 ⊢ (𝑦 = ^◡𝑥 → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ↔ ( I ↾ (dom ^◡𝑥 ∪ ran ^◡𝑥)) ⊆ ^◡𝑥))
60	53, 59	imbitrrid 246	. . 3 ⊢ (𝑦 = ^◡𝑥 → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))
61	60	adantl 481	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 = ^◡𝑥) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))
62		dmeq 5917	. . . . 5 ⊢ (𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) → dom 𝑥 = dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
63		rneq 5950	. . . . 5 ⊢ (𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) → ran 𝑥 = ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
64	62, 63	uneq12d 4179	. . . 4 ⊢ (𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) → (dom 𝑥 ∪ ran 𝑥) = (dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∪ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋)))))
65	64	reseq2d 6000	. . 3 ⊢ (𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∪ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))))
66		id 22	. . 3 ⊢ (𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) → 𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
67	65, 66	sseq12d 4029	. 2 ⊢ (𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 ↔ ( I ↾ (dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∪ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))) ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋)))))
68		ssun1 4188	. . 3 ⊢ 𝑋 ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
69	68	a1i 11	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
70		dmexg 7924	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → dom 𝑋 ∈ V)
71		rnexg 7925	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ran 𝑋 ∈ V)
72	70, 71	unexd 7773	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (dom 𝑋 ∪ ran 𝑋) ∈ V)
73	72	resiexd 7236	. . 3 ⊢ (𝑋 ∈ 𝑉 → ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ∈ V)
74		unexg 7762	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ∈ V) → (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∈ V)
75	73, 74	mpdan 687	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∈ V)
76		dmun 5924	. . . . . 6 ⊢ dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) = (dom 𝑋 ∪ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
77		ssun1 4188	. . . . . . 7 ⊢ dom 𝑋 ⊆ (dom 𝑋 ∪ ran 𝑋)
78		resdmss 6257	. . . . . . 7 ⊢ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ⊆ (dom 𝑋 ∪ ran 𝑋)
79	77, 78	unssi 4201	. . . . . 6 ⊢ (dom 𝑋 ∪ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
80	76, 79	eqsstri 4030	. . . . 5 ⊢ dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
81		rnun 6168	. . . . . 6 ⊢ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) = (ran 𝑋 ∪ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
82		ssun2 4189	. . . . . . 7 ⊢ ran 𝑋 ⊆ (dom 𝑋 ∪ ran 𝑋)
83		rnresi 6095	. . . . . . . 8 ⊢ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)) = (dom 𝑋 ∪ ran 𝑋)
84	83	eqimssi 4056	. . . . . . 7 ⊢ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ⊆ (dom 𝑋 ∪ ran 𝑋)
85	82, 84	unssi 4201	. . . . . 6 ⊢ (ran 𝑋 ∪ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
86	81, 85	eqsstri 4030	. . . . 5 ⊢ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
87	80, 86	pm3.2i 470	. . . 4 ⊢ (dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋) ∧ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋))
88		unss 4200	. . . . 5 ⊢ ((dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋) ∧ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)) ↔ (dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∪ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋)))) ⊆ (dom 𝑋 ∪ ran 𝑋))
89		ssres2 6025	. . . . 5 ⊢ ((dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∪ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋)))) ⊆ (dom 𝑋 ∪ ran 𝑋) → ( I ↾ (dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∪ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))) ⊆ ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
90	88, 89	sylbi 217	. . . 4 ⊢ ((dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋) ∧ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)) → ( I ↾ (dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∪ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))) ⊆ ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
91		ssun4 4191	. . . 4 ⊢ (( I ↾ (dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∪ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))) ⊆ ( I ↾ (dom 𝑋 ∪ ran 𝑋)) → ( I ↾ (dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∪ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))) ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
92	87, 90, 91	mp2b 10	. . 3 ⊢ ( I ↾ (dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∪ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))) ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
93	92	a1i 11	. 2 ⊢ (𝑋 ∈ 𝑉 → ( I ↾ (dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∪ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))) ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
94	44, 61, 67, 69, 75, 93	clcnvlem 43613	1 ⊢ (𝑋 ∈ 𝑉 → ^◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} = ∩ {𝑦 ∣ (^◡𝑋 ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {cab 2712 Vcvv 3478 ∖ cdif 3960 ∪ cun 3961 ⊆ wss 3963 ∅c0 4339 ∩ cint 4951 I cid 5582 ^◡ccnv 5688 dom cdm 5689 ran crn 5690 ↾ cres 5691

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-1st 8013 df-2nd 8014

This theorem is referenced by: (None)

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