Theorem cnvrcl0 39978

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 6427	. . . . . . 7 ⊢ ◡( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom 𝑦 ∪ ran 𝑦))
2		cnvnonrel 39941	. . . . . . . . . . . . . . . 16 ⊢ ◡(𝑋 ∖ ◡◡𝑋) = ∅
3		cnv0 5993	. . . . . . . . . . . . . . . 16 ⊢ ◡∅ = ∅
4	2, 3	eqtr4i 2847	. . . . . . . . . . . . . . 15 ⊢ ◡(𝑋 ∖ ◡◡𝑋) = ◡∅
5	4	dmeqi 5767	. . . . . . . . . . . . . 14 ⊢ dom ◡(𝑋 ∖ ◡◡𝑋) = dom ◡∅
6		df-rn 5560	. . . . . . . . . . . . . 14 ⊢ ran (𝑋 ∖ ◡◡𝑋) = dom ◡(𝑋 ∖ ◡◡𝑋)
7		df-rn 5560	. . . . . . . . . . . . . 14 ⊢ ran ∅ = dom ◡∅
8	5, 6, 7	3eqtr4i 2854	. . . . . . . . . . . . 13 ⊢ ran (𝑋 ∖ ◡◡𝑋) = ran ∅
9		0ss 4349	. . . . . . . . . . . . . 14 ⊢ ∅ ⊆ ◡𝑦
10	9	rnssi 5804	. . . . . . . . . . . . 13 ⊢ ran ∅ ⊆ ran ◡𝑦
11	8, 10	eqsstri 4000	. . . . . . . . . . . 12 ⊢ ran (𝑋 ∖ ◡◡𝑋) ⊆ ran ◡𝑦
12		ssequn2 4158	. . . . . . . . . . . 12 ⊢ (ran (𝑋 ∖ ◡◡𝑋) ⊆ ran ◡𝑦 ↔ (ran ◡𝑦 ∪ ran (𝑋 ∖ ◡◡𝑋)) = ran ◡𝑦)
13	11, 12	mpbi 232	. . . . . . . . . . 11 ⊢ (ran ◡𝑦 ∪ ran (𝑋 ∖ ◡◡𝑋)) = ran ◡𝑦
14		rnun 5998	. . . . . . . . . . 11 ⊢ ran (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) = (ran ◡𝑦 ∪ ran (𝑋 ∖ ◡◡𝑋))
15		dfdm4 5758	. . . . . . . . . . 11 ⊢ dom 𝑦 = ran ◡𝑦
16	13, 14, 15	3eqtr4ri 2855	. . . . . . . . . 10 ⊢ dom 𝑦 = ran (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))
17	4	rneqi 5801	. . . . . . . . . . . . . 14 ⊢ ran ◡(𝑋 ∖ ◡◡𝑋) = ran ◡∅
18		dfdm4 5758	. . . . . . . . . . . . . 14 ⊢ dom (𝑋 ∖ ◡◡𝑋) = ran ◡(𝑋 ∖ ◡◡𝑋)
19		dfdm4 5758	. . . . . . . . . . . . . 14 ⊢ dom ∅ = ran ◡∅
20	17, 18, 19	3eqtr4i 2854	. . . . . . . . . . . . 13 ⊢ dom (𝑋 ∖ ◡◡𝑋) = dom ∅
21		dmss 5765	. . . . . . . . . . . . . 14 ⊢ (∅ ⊆ ◡𝑦 → dom ∅ ⊆ dom ◡𝑦)
22	9, 21	ax-mp 5	. . . . . . . . . . . . 13 ⊢ dom ∅ ⊆ dom ◡𝑦
23	20, 22	eqsstri 4000	. . . . . . . . . . . 12 ⊢ dom (𝑋 ∖ ◡◡𝑋) ⊆ dom ◡𝑦
24		ssequn2 4158	. . . . . . . . . . . 12 ⊢ (dom (𝑋 ∖ ◡◡𝑋) ⊆ dom ◡𝑦 ↔ (dom ◡𝑦 ∪ dom (𝑋 ∖ ◡◡𝑋)) = dom ◡𝑦)
25	23, 24	mpbi 232	. . . . . . . . . . 11 ⊢ (dom ◡𝑦 ∪ dom (𝑋 ∖ ◡◡𝑋)) = dom ◡𝑦
26		dmun 5773	. . . . . . . . . . 11 ⊢ dom (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) = (dom ◡𝑦 ∪ dom (𝑋 ∖ ◡◡𝑋))
27		df-rn 5560	. . . . . . . . . . 11 ⊢ ran 𝑦 = dom ◡𝑦
28	25, 26, 27	3eqtr4ri 2855	. . . . . . . . . 10 ⊢ ran 𝑦 = dom (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))
29	16, 28	uneq12i 4136	. . . . . . . . 9 ⊢ (dom 𝑦 ∪ ran 𝑦) = (ran (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) ∪ dom (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)))
30	29	equncomi 4130	. . . . . . . 8 ⊢ (dom 𝑦 ∪ ran 𝑦) = (dom (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) ∪ ran (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)))
31	30	reseq2i 5844	. . . . . . 7 ⊢ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) ∪ ran (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))))
32	1, 31	eqtr2i 2845	. . . . . 6 ⊢ ( I ↾ (dom (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) ∪ ran (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)))) = ◡( I ↾ (dom 𝑦 ∪ ran 𝑦))
33		cnvss 5737	. . . . . 6 ⊢ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ◡( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ ◡𝑦)
34	32, 33	eqsstrid 4014	. . . . 5 ⊢ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) ∪ ran (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)))) ⊆ ◡𝑦)
35		ssun1 4147	. . . . 5 ⊢ ◡𝑦 ⊆ (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))
36	34, 35	sstrdi 3978	. . . 4 ⊢ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) ∪ ran (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)))) ⊆ (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)))
37		dmeq 5766	. . . . . . 7 ⊢ (𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) → dom 𝑥 = dom (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)))
38		rneq 5800	. . . . . . 7 ⊢ (𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) → ran 𝑥 = ran (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)))
39	37, 38	uneq12d 4139	. . . . . 6 ⊢ (𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) → (dom 𝑥 ∪ ran 𝑥) = (dom (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) ∪ ran (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))))
40	39	reseq2d 5847	. . . . 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 3999	. . . 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	syl5ibr 248	. . 3 ⊢ (𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))
44	43	adantl 484	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))) → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))
45		cnvresid 6427	. . . . . 6 ⊢ ◡( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑥 ∪ ran 𝑥))
46		dfdm4 5758	. . . . . . . . 9 ⊢ dom 𝑥 = ran ◡𝑥
47		df-rn 5560	. . . . . . . . 9 ⊢ ran 𝑥 = dom ◡𝑥
48	46, 47	uneq12i 4136	. . . . . . . 8 ⊢ (dom 𝑥 ∪ ran 𝑥) = (ran ◡𝑥 ∪ dom ◡𝑥)
49	48	equncomi 4130	. . . . . . 7 ⊢ (dom 𝑥 ∪ ran 𝑥) = (dom ◡𝑥 ∪ ran ◡𝑥)
50	49	reseq2i 5844	. . . . . 6 ⊢ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom ◡𝑥 ∪ ran ◡𝑥))
51	45, 50	eqtr2i 2845	. . . . 5 ⊢ ( I ↾ (dom ◡𝑥 ∪ ran ◡𝑥)) = ◡( I ↾ (dom 𝑥 ∪ ran 𝑥))
52		cnvss 5737	. . . . 5 ⊢ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ◡( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ ◡𝑥)
53	51, 52	eqsstrid 4014	. . . 4 ⊢ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom ◡𝑥 ∪ ran ◡𝑥)) ⊆ ◡𝑥)
54		dmeq 5766	. . . . . . 7 ⊢ (𝑦 = ◡𝑥 → dom 𝑦 = dom ◡𝑥)
55		rneq 5800	. . . . . . 7 ⊢ (𝑦 = ◡𝑥 → ran 𝑦 = ran ◡𝑥)
56	54, 55	uneq12d 4139	. . . . . 6 ⊢ (𝑦 = ◡𝑥 → (dom 𝑦 ∪ ran 𝑦) = (dom ◡𝑥 ∪ ran ◡𝑥))
57	56	reseq2d 5847	. . . . 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 3999	. . . 4 ⊢ (𝑦 = ◡𝑥 → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ↔ ( I ↾ (dom ◡𝑥 ∪ ran ◡𝑥)) ⊆ ◡𝑥))
60	53, 59	syl5ibr 248	. . 3 ⊢ (𝑦 = ◡𝑥 → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))
61	60	adantl 484	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 = ◡𝑥) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))
62		dmeq 5766	. . . . 5 ⊢ (𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) → dom 𝑥 = dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
63		rneq 5800	. . . . 5 ⊢ (𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) → ran 𝑥 = ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
64	62, 63	uneq12d 4139	. . . 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 5847	. . 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 3999	. 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 4147	. . 3 ⊢ 𝑋 ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
69	68	a1i 11	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
70		dmexg 7607	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → dom 𝑋 ∈ V)
71		rnexg 7608	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ran 𝑋 ∈ V)
72		unexg 7466	. . . . 5 ⊢ ((dom 𝑋 ∈ V ∧ ran 𝑋 ∈ V) → (dom 𝑋 ∪ ran 𝑋) ∈ V)
73	70, 71, 72	syl2anc 586	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (dom 𝑋 ∪ ran 𝑋) ∈ V)
74	73	resiexd 6973	. . 3 ⊢ (𝑋 ∈ 𝑉 → ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ∈ V)
75		unexg 7466	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ∈ V) → (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∈ V)
76	74, 75	mpdan 685	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∈ V)
77		dmun 5773	. . . . . 6 ⊢ dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) = (dom 𝑋 ∪ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
78		ssun1 4147	. . . . . . 7 ⊢ dom 𝑋 ⊆ (dom 𝑋 ∪ ran 𝑋)
79		dmresi 5915	. . . . . . . 8 ⊢ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋)) = (dom 𝑋 ∪ ran 𝑋)
80	79	eqimssi 4024	. . . . . . 7 ⊢ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ⊆ (dom 𝑋 ∪ ran 𝑋)
81	78, 80	unssi 4160	. . . . . 6 ⊢ (dom 𝑋 ∪ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
82	77, 81	eqsstri 4000	. . . . 5 ⊢ dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
83		rnun 5998	. . . . . 6 ⊢ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) = (ran 𝑋 ∪ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
84		ssun2 4148	. . . . . . 7 ⊢ ran 𝑋 ⊆ (dom 𝑋 ∪ ran 𝑋)
85		rnresi 5937	. . . . . . . 8 ⊢ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)) = (dom 𝑋 ∪ ran 𝑋)
86	85	eqimssi 4024	. . . . . . 7 ⊢ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ⊆ (dom 𝑋 ∪ ran 𝑋)
87	84, 86	unssi 4160	. . . . . 6 ⊢ (ran 𝑋 ∪ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
88	83, 87	eqsstri 4000	. . . . 5 ⊢ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
89	82, 88	pm3.2i 473	. . . 4 ⊢ (dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋) ∧ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋))
90		unss 4159	. . . . 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))
91		ssres2 5875	. . . . 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)))
92	90, 91	sylbi 219	. . . 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)))
93		ssun4 4150	. . . 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))))
94	89, 92, 93	mp2b 10	. . 3 ⊢ ( I ↾ (dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∪ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))) ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
95	94	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))))
96	44, 61, 67, 69, 76, 95	clcnvlem 39976	1 ⊢ (𝑋 ∈ 𝑉 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)})

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  {cab 2799  Vcvv 3494   ∖ cdif 3932   ∪ cun 3933   ⊆ wss 3935  ∅c0 4290  ∩ cint 4868   I cid 5453  ^◡ccnv 5548  dom cdm 5549  ran crn 5550   ↾ cres 5551

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-1st 7683  df-2nd 7684

This theorem is referenced by: (None)