cnvrcl0 - Mathbox for Richard Penner

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Theorem cnvrcl0 40325

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 6403	. . . . . . 7 ⊢ ^◡( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom 𝑦 ∪ ran 𝑦))
2		cnvnonrel 40288	. . . . . . . . . . . . . . . 16 ⊢ ^◡(𝑋 ∖ ^◡^◡𝑋) = ∅
3		cnv0 5966	. . . . . . . . . . . . . . . 16 ⊢ ^◡∅ = ∅
4	2, 3	eqtr4i 2824	. . . . . . . . . . . . . . 15 ⊢ ^◡(𝑋 ∖ ^◡^◡𝑋) = ^◡∅
5	4	dmeqi 5737	. . . . . . . . . . . . . 14 ⊢ dom ^◡(𝑋 ∖ ^◡^◡𝑋) = dom ^◡∅
6		df-rn 5530	. . . . . . . . . . . . . 14 ⊢ ran (𝑋 ∖ ^◡^◡𝑋) = dom ^◡(𝑋 ∖ ^◡^◡𝑋)
7		df-rn 5530	. . . . . . . . . . . . . 14 ⊢ ran ∅ = dom ^◡∅
8	5, 6, 7	3eqtr4i 2831	. . . . . . . . . . . . 13 ⊢ ran (𝑋 ∖ ^◡^◡𝑋) = ran ∅
9		0ss 4304	. . . . . . . . . . . . . 14 ⊢ ∅ ⊆ ^◡𝑦
10	9	rnssi 5774	. . . . . . . . . . . . 13 ⊢ ran ∅ ⊆ ran ^◡𝑦
11	8, 10	eqsstri 3949	. . . . . . . . . . . 12 ⊢ ran (𝑋 ∖ ^◡^◡𝑋) ⊆ ran ^◡𝑦
12		ssequn2 4110	. . . . . . . . . . . 12 ⊢ (ran (𝑋 ∖ ^◡^◡𝑋) ⊆ ran ^◡𝑦 ↔ (ran ^◡𝑦 ∪ ran (𝑋 ∖ ^◡^◡𝑋)) = ran ^◡𝑦)
13	11, 12	mpbi 233	. . . . . . . . . . 11 ⊢ (ran ^◡𝑦 ∪ ran (𝑋 ∖ ^◡^◡𝑋)) = ran ^◡𝑦
14		rnun 5971	. . . . . . . . . . 11 ⊢ ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) = (ran ^◡𝑦 ∪ ran (𝑋 ∖ ^◡^◡𝑋))
15		dfdm4 5728	. . . . . . . . . . 11 ⊢ dom 𝑦 = ran ^◡𝑦
16	13, 14, 15	3eqtr4ri 2832	. . . . . . . . . 10 ⊢ dom 𝑦 = ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))
17	4	rneqi 5771	. . . . . . . . . . . . . 14 ⊢ ran ^◡(𝑋 ∖ ^◡^◡𝑋) = ran ^◡∅
18		dfdm4 5728	. . . . . . . . . . . . . 14 ⊢ dom (𝑋 ∖ ^◡^◡𝑋) = ran ^◡(𝑋 ∖ ^◡^◡𝑋)
19		dfdm4 5728	. . . . . . . . . . . . . 14 ⊢ dom ∅ = ran ^◡∅
20	17, 18, 19	3eqtr4i 2831	. . . . . . . . . . . . 13 ⊢ dom (𝑋 ∖ ^◡^◡𝑋) = dom ∅
21		dmss 5735	. . . . . . . . . . . . . 14 ⊢ (∅ ⊆ ^◡𝑦 → dom ∅ ⊆ dom ^◡𝑦)
22	9, 21	ax-mp 5	. . . . . . . . . . . . 13 ⊢ dom ∅ ⊆ dom ^◡𝑦
23	20, 22	eqsstri 3949	. . . . . . . . . . . 12 ⊢ dom (𝑋 ∖ ^◡^◡𝑋) ⊆ dom ^◡𝑦
24		ssequn2 4110	. . . . . . . . . . . 12 ⊢ (dom (𝑋 ∖ ^◡^◡𝑋) ⊆ dom ^◡𝑦 ↔ (dom ^◡𝑦 ∪ dom (𝑋 ∖ ^◡^◡𝑋)) = dom ^◡𝑦)
25	23, 24	mpbi 233	. . . . . . . . . . 11 ⊢ (dom ^◡𝑦 ∪ dom (𝑋 ∖ ^◡^◡𝑋)) = dom ^◡𝑦
26		dmun 5743	. . . . . . . . . . 11 ⊢ dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) = (dom ^◡𝑦 ∪ dom (𝑋 ∖ ^◡^◡𝑋))
27		df-rn 5530	. . . . . . . . . . 11 ⊢ ran 𝑦 = dom ^◡𝑦
28	25, 26, 27	3eqtr4ri 2832	. . . . . . . . . 10 ⊢ ran 𝑦 = dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))
29	16, 28	uneq12i 4088	. . . . . . . . 9 ⊢ (dom 𝑦 ∪ ran 𝑦) = (ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∪ dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))
30	29	equncomi 4082	. . . . . . . 8 ⊢ (dom 𝑦 ∪ ran 𝑦) = (dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∪ ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))
31	30	reseq2i 5815	. . . . . . 7 ⊢ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∪ ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))))
32	1, 31	eqtr2i 2822	. . . . . 6 ⊢ ( I ↾ (dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∪ ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))) = ^◡( I ↾ (dom 𝑦 ∪ ran 𝑦))
33		cnvss 5707	. . . . . 6 ⊢ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ^◡( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ ^◡𝑦)
34	32, 33	eqsstrid 3963	. . . . 5 ⊢ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∪ ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))) ⊆ ^◡𝑦)
35		ssun1 4099	. . . . 5 ⊢ ^◡𝑦 ⊆ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))
36	34, 35	sstrdi 3927	. . . 4 ⊢ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∪ ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))) ⊆ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))
37		dmeq 5736	. . . . . . 7 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → dom 𝑥 = dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))
38		rneq 5770	. . . . . . 7 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → ran 𝑥 = ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))
39	37, 38	uneq12d 4091	. . . . . 6 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → (dom 𝑥 ∪ ran 𝑥) = (dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∪ ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))))
40	39	reseq2d 5818	. . . . 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 3948	. . . 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 249	. . 3 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))
44	43	adantl 485	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))
45		cnvresid 6403	. . . . . 6 ⊢ ^◡( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑥 ∪ ran 𝑥))
46		dfdm4 5728	. . . . . . . . 9 ⊢ dom 𝑥 = ran ^◡𝑥
47		df-rn 5530	. . . . . . . . 9 ⊢ ran 𝑥 = dom ^◡𝑥
48	46, 47	uneq12i 4088	. . . . . . . 8 ⊢ (dom 𝑥 ∪ ran 𝑥) = (ran ^◡𝑥 ∪ dom ^◡𝑥)
49	48	equncomi 4082	. . . . . . 7 ⊢ (dom 𝑥 ∪ ran 𝑥) = (dom ^◡𝑥 ∪ ran ^◡𝑥)
50	49	reseq2i 5815	. . . . . 6 ⊢ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom ^◡𝑥 ∪ ran ^◡𝑥))
51	45, 50	eqtr2i 2822	. . . . 5 ⊢ ( I ↾ (dom ^◡𝑥 ∪ ran ^◡𝑥)) = ^◡( I ↾ (dom 𝑥 ∪ ran 𝑥))
52		cnvss 5707	. . . . 5 ⊢ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ^◡( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ ^◡𝑥)
53	51, 52	eqsstrid 3963	. . . 4 ⊢ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom ^◡𝑥 ∪ ran ^◡𝑥)) ⊆ ^◡𝑥)
54		dmeq 5736	. . . . . . 7 ⊢ (𝑦 = ^◡𝑥 → dom 𝑦 = dom ^◡𝑥)
55		rneq 5770	. . . . . . 7 ⊢ (𝑦 = ^◡𝑥 → ran 𝑦 = ran ^◡𝑥)
56	54, 55	uneq12d 4091	. . . . . 6 ⊢ (𝑦 = ^◡𝑥 → (dom 𝑦 ∪ ran 𝑦) = (dom ^◡𝑥 ∪ ran ^◡𝑥))
57	56	reseq2d 5818	. . . . 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 3948	. . . 4 ⊢ (𝑦 = ^◡𝑥 → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ↔ ( I ↾ (dom ^◡𝑥 ∪ ran ^◡𝑥)) ⊆ ^◡𝑥))
60	53, 59	syl5ibr 249	. . 3 ⊢ (𝑦 = ^◡𝑥 → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))
61	60	adantl 485	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 = ^◡𝑥) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))
62		dmeq 5736	. . . . 5 ⊢ (𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) → dom 𝑥 = dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
63		rneq 5770	. . . . 5 ⊢ (𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) → ran 𝑥 = ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
64	62, 63	uneq12d 4091	. . . 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 5818	. . 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 3948	. 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 4099	. . 3 ⊢ 𝑋 ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
69	68	a1i 11	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
70		dmexg 7594	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → dom 𝑋 ∈ V)
71		rnexg 7595	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ran 𝑋 ∈ V)
72		unexg 7452	. . . . 5 ⊢ ((dom 𝑋 ∈ V ∧ ran 𝑋 ∈ V) → (dom 𝑋 ∪ ran 𝑋) ∈ V)
73	70, 71, 72	syl2anc 587	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (dom 𝑋 ∪ ran 𝑋) ∈ V)
74	73	resiexd 6956	. . 3 ⊢ (𝑋 ∈ 𝑉 → ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ∈ V)
75		unexg 7452	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ∈ V) → (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∈ V)
76	74, 75	mpdan 686	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∈ V)
77		dmun 5743	. . . . . 6 ⊢ dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) = (dom 𝑋 ∪ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
78		ssun1 4099	. . . . . . 7 ⊢ dom 𝑋 ⊆ (dom 𝑋 ∪ ran 𝑋)
79		dmresi 5888	. . . . . . . 8 ⊢ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋)) = (dom 𝑋 ∪ ran 𝑋)
80	79	eqimssi 3973	. . . . . . 7 ⊢ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ⊆ (dom 𝑋 ∪ ran 𝑋)
81	78, 80	unssi 4112	. . . . . 6 ⊢ (dom 𝑋 ∪ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
82	77, 81	eqsstri 3949	. . . . 5 ⊢ dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
83		rnun 5971	. . . . . 6 ⊢ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) = (ran 𝑋 ∪ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
84		ssun2 4100	. . . . . . 7 ⊢ ran 𝑋 ⊆ (dom 𝑋 ∪ ran 𝑋)
85		rnresi 5910	. . . . . . . 8 ⊢ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)) = (dom 𝑋 ∪ ran 𝑋)
86	85	eqimssi 3973	. . . . . . 7 ⊢ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ⊆ (dom 𝑋 ∪ ran 𝑋)
87	84, 86	unssi 4112	. . . . . 6 ⊢ (ran 𝑋 ∪ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
88	83, 87	eqsstri 3949	. . . . 5 ⊢ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
89	82, 88	pm3.2i 474	. . . 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 4111	. . . . 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 5846	. . . . 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 220	. . . 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 4102	. . . 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 40323	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 399 = wceq 1538 ∈ wcel 2111 {cab 2776 Vcvv 3441 ∖ cdif 3878 ∪ cun 3879 ⊆ wss 3881 ∅c0 4243 ∩ cint 4838 I cid 5424 ^◡ccnv 5518 dom cdm 5519 ran crn 5520 ↾ cres 5521

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-1st 7671 df-2nd 7672

This theorem is referenced by: (None)

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