Theorem cnvrcl0 37932

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 5968	. . . . . . 7 ⊢ ◡( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom 𝑦 ∪ ran 𝑦))
2		cnvnonrel 37894	. . . . . . . . . . . . . . . 16 ⊢ ◡(𝑋 ∖ ◡◡𝑋) = ∅
3		cnv0 5535	. . . . . . . . . . . . . . . 16 ⊢ ◡∅ = ∅
4	2, 3	eqtr4i 2647	. . . . . . . . . . . . . . 15 ⊢ ◡(𝑋 ∖ ◡◡𝑋) = ◡∅
5	4	dmeqi 5325	. . . . . . . . . . . . . 14 ⊢ dom ◡(𝑋 ∖ ◡◡𝑋) = dom ◡∅
6		df-rn 5125	. . . . . . . . . . . . . 14 ⊢ ran (𝑋 ∖ ◡◡𝑋) = dom ◡(𝑋 ∖ ◡◡𝑋)
7		df-rn 5125	. . . . . . . . . . . . . 14 ⊢ ran ∅ = dom ◡∅
8	5, 6, 7	3eqtr4i 2654	. . . . . . . . . . . . 13 ⊢ ran (𝑋 ∖ ◡◡𝑋) = ran ∅
9		0ss 3972	. . . . . . . . . . . . . 14 ⊢ ∅ ⊆ ◡𝑦
10		rnss 5354	. . . . . . . . . . . . . 14 ⊢ (∅ ⊆ ◡𝑦 → ran ∅ ⊆ ran ◡𝑦)
11	9, 10	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ran ∅ ⊆ ran ◡𝑦
12	8, 11	eqsstri 3635	. . . . . . . . . . . 12 ⊢ ran (𝑋 ∖ ◡◡𝑋) ⊆ ran ◡𝑦
13		ssequn2 3786	. . . . . . . . . . . 12 ⊢ (ran (𝑋 ∖ ◡◡𝑋) ⊆ ran ◡𝑦 ↔ (ran ◡𝑦 ∪ ran (𝑋 ∖ ◡◡𝑋)) = ran ◡𝑦)
14	12, 13	mpbi 220	. . . . . . . . . . 11 ⊢ (ran ◡𝑦 ∪ ran (𝑋 ∖ ◡◡𝑋)) = ran ◡𝑦
15		rnun 5541	. . . . . . . . . . 11 ⊢ ran (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) = (ran ◡𝑦 ∪ ran (𝑋 ∖ ◡◡𝑋))
16		dfdm4 5316	. . . . . . . . . . 11 ⊢ dom 𝑦 = ran ◡𝑦
17	14, 15, 16	3eqtr4ri 2655	. . . . . . . . . 10 ⊢ dom 𝑦 = ran (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))
18	4	rneqi 5352	. . . . . . . . . . . . . 14 ⊢ ran ◡(𝑋 ∖ ◡◡𝑋) = ran ◡∅
19		dfdm4 5316	. . . . . . . . . . . . . 14 ⊢ dom (𝑋 ∖ ◡◡𝑋) = ran ◡(𝑋 ∖ ◡◡𝑋)
20		dfdm4 5316	. . . . . . . . . . . . . 14 ⊢ dom ∅ = ran ◡∅
21	18, 19, 20	3eqtr4i 2654	. . . . . . . . . . . . 13 ⊢ dom (𝑋 ∖ ◡◡𝑋) = dom ∅
22		dmss 5323	. . . . . . . . . . . . . 14 ⊢ (∅ ⊆ ◡𝑦 → dom ∅ ⊆ dom ◡𝑦)
23	9, 22	ax-mp 5	. . . . . . . . . . . . 13 ⊢ dom ∅ ⊆ dom ◡𝑦
24	21, 23	eqsstri 3635	. . . . . . . . . . . 12 ⊢ dom (𝑋 ∖ ◡◡𝑋) ⊆ dom ◡𝑦
25		ssequn2 3786	. . . . . . . . . . . 12 ⊢ (dom (𝑋 ∖ ◡◡𝑋) ⊆ dom ◡𝑦 ↔ (dom ◡𝑦 ∪ dom (𝑋 ∖ ◡◡𝑋)) = dom ◡𝑦)
26	24, 25	mpbi 220	. . . . . . . . . . 11 ⊢ (dom ◡𝑦 ∪ dom (𝑋 ∖ ◡◡𝑋)) = dom ◡𝑦
27		dmun 5331	. . . . . . . . . . 11 ⊢ dom (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) = (dom ◡𝑦 ∪ dom (𝑋 ∖ ◡◡𝑋))
28		df-rn 5125	. . . . . . . . . . 11 ⊢ ran 𝑦 = dom ◡𝑦
29	26, 27, 28	3eqtr4ri 2655	. . . . . . . . . 10 ⊢ ran 𝑦 = dom (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))
30	17, 29	uneq12i 3765	. . . . . . . . 9 ⊢ (dom 𝑦 ∪ ran 𝑦) = (ran (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) ∪ dom (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)))
31	30	equncomi 3759	. . . . . . . 8 ⊢ (dom 𝑦 ∪ ran 𝑦) = (dom (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) ∪ ran (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)))
32	31	reseq2i 5393	. . . . . . 7 ⊢ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) ∪ ran (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))))
33	1, 32	eqtr2i 2645	. . . . . 6 ⊢ ( I ↾ (dom (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) ∪ ran (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)))) = ◡( I ↾ (dom 𝑦 ∪ ran 𝑦))
34		cnvss 5294	. . . . . 6 ⊢ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ◡( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ ◡𝑦)
35	33, 34	syl5eqss 3649	. . . . 5 ⊢ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) ∪ ran (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)))) ⊆ ◡𝑦)
36		ssun1 3776	. . . . 5 ⊢ ◡𝑦 ⊆ (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))
37	35, 36	syl6ss 3615	. . . 4 ⊢ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) ∪ ran (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)))) ⊆ (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)))
38		dmeq 5324	. . . . . . 7 ⊢ (𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) → dom 𝑥 = dom (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)))
39		rneq 5351	. . . . . . 7 ⊢ (𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) → ran 𝑥 = ran (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)))
40	38, 39	uneq12d 3768	. . . . . 6 ⊢ (𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) → (dom 𝑥 ∪ ran 𝑥) = (dom (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) ∪ ran (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))))
41	40	reseq2d 5396	. . . . 5 ⊢ (𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) ∪ ran (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)))))
42		id 22	. . . . 5 ⊢ (𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) → 𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)))
43	41, 42	sseq12d 3634	. . . 4 ⊢ (𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 ↔ ( I ↾ (dom (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) ∪ ran (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)))) ⊆ (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))))
44	37, 43	syl5ibr 236	. . 3 ⊢ (𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))
45	44	adantl 482	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))) → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))
46		cnvresid 5968	. . . . . 6 ⊢ ◡( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑥 ∪ ran 𝑥))
47		dfdm4 5316	. . . . . . . . 9 ⊢ dom 𝑥 = ran ◡𝑥
48		df-rn 5125	. . . . . . . . 9 ⊢ ran 𝑥 = dom ◡𝑥
49	47, 48	uneq12i 3765	. . . . . . . 8 ⊢ (dom 𝑥 ∪ ran 𝑥) = (ran ◡𝑥 ∪ dom ◡𝑥)
50	49	equncomi 3759	. . . . . . 7 ⊢ (dom 𝑥 ∪ ran 𝑥) = (dom ◡𝑥 ∪ ran ◡𝑥)
51	50	reseq2i 5393	. . . . . 6 ⊢ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom ◡𝑥 ∪ ran ◡𝑥))
52	46, 51	eqtr2i 2645	. . . . 5 ⊢ ( I ↾ (dom ◡𝑥 ∪ ran ◡𝑥)) = ◡( I ↾ (dom 𝑥 ∪ ran 𝑥))
53		cnvss 5294	. . . . 5 ⊢ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ◡( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ ◡𝑥)
54	52, 53	syl5eqss 3649	. . . 4 ⊢ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom ◡𝑥 ∪ ran ◡𝑥)) ⊆ ◡𝑥)
55		dmeq 5324	. . . . . . 7 ⊢ (𝑦 = ◡𝑥 → dom 𝑦 = dom ◡𝑥)
56		rneq 5351	. . . . . . 7 ⊢ (𝑦 = ◡𝑥 → ran 𝑦 = ran ◡𝑥)
57	55, 56	uneq12d 3768	. . . . . 6 ⊢ (𝑦 = ◡𝑥 → (dom 𝑦 ∪ ran 𝑦) = (dom ◡𝑥 ∪ ran ◡𝑥))
58	57	reseq2d 5396	. . . . 5 ⊢ (𝑦 = ◡𝑥 → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom ◡𝑥 ∪ ran ◡𝑥)))
59		id 22	. . . . 5 ⊢ (𝑦 = ◡𝑥 → 𝑦 = ◡𝑥)
60	58, 59	sseq12d 3634	. . . 4 ⊢ (𝑦 = ◡𝑥 → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ↔ ( I ↾ (dom ◡𝑥 ∪ ran ◡𝑥)) ⊆ ◡𝑥))
61	54, 60	syl5ibr 236	. . 3 ⊢ (𝑦 = ◡𝑥 → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))
62	61	adantl 482	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 = ◡𝑥) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))
63		dmeq 5324	. . . . 5 ⊢ (𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) → dom 𝑥 = dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
64		rneq 5351	. . . . 5 ⊢ (𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) → ran 𝑥 = ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
65	63, 64	uneq12d 3768	. . . 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)))))
66	65	reseq2d 5396	. . 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))))))
67		id 22	. . 3 ⊢ (𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) → 𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
68	66, 67	sseq12d 3634	. 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)))))
69		ssun1 3776	. . 3 ⊢ 𝑋 ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
70	69	a1i 11	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
71		dmexg 7097	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → dom 𝑋 ∈ V)
72		rnexg 7098	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ran 𝑋 ∈ V)
73		unexg 6959	. . . . 5 ⊢ ((dom 𝑋 ∈ V ∧ ran 𝑋 ∈ V) → (dom 𝑋 ∪ ran 𝑋) ∈ V)
74	71, 72, 73	syl2anc 693	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (dom 𝑋 ∪ ran 𝑋) ∈ V)
75	74	resiexd 6480	. . 3 ⊢ (𝑋 ∈ 𝑉 → ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ∈ V)
76		unexg 6959	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ∈ V) → (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∈ V)
77	75, 76	mpdan 702	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∈ V)
78		dmun 5331	. . . . . 6 ⊢ dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) = (dom 𝑋 ∪ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
79		ssun1 3776	. . . . . . 7 ⊢ dom 𝑋 ⊆ (dom 𝑋 ∪ ran 𝑋)
80		dmresi 5457	. . . . . . . 8 ⊢ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋)) = (dom 𝑋 ∪ ran 𝑋)
81	80	eqimssi 3659	. . . . . . 7 ⊢ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ⊆ (dom 𝑋 ∪ ran 𝑋)
82	79, 81	unssi 3788	. . . . . 6 ⊢ (dom 𝑋 ∪ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
83	78, 82	eqsstri 3635	. . . . 5 ⊢ dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
84		rnun 5541	. . . . . 6 ⊢ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) = (ran 𝑋 ∪ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
85		ssun2 3777	. . . . . . 7 ⊢ ran 𝑋 ⊆ (dom 𝑋 ∪ ran 𝑋)
86		rnresi 5479	. . . . . . . 8 ⊢ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)) = (dom 𝑋 ∪ ran 𝑋)
87	86	eqimssi 3659	. . . . . . 7 ⊢ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ⊆ (dom 𝑋 ∪ ran 𝑋)
88	85, 87	unssi 3788	. . . . . 6 ⊢ (ran 𝑋 ∪ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
89	84, 88	eqsstri 3635	. . . . 5 ⊢ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
90	83, 89	pm3.2i 471	. . . 4 ⊢ (dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋) ∧ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋))
91		unss 3787	. . . . 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))
92		ssres2 5425	. . . . 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)))
93	91, 92	sylbi 207	. . . 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)))
94		ssun4 3779	. . . 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))))
95	90, 93, 94	mp2b 10	. . 3 ⊢ ( I ↾ (dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∪ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))) ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
96	95	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))))
97	45, 62, 68, 70, 77, 96	clcnvlem 37930	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 384   = wceq 1483   ∈ wcel 1990  {cab 2608  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  ∩ cint 4475   I cid 5023  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   ↾ cres 5116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by: (None)