cnvrcl0 - Mathbox for Richard Penner

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Theorem cnvrcl0 44411

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 6619	. . . . . . 7 ⊢ ^◡( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom 𝑦 ∪ ran 𝑦))
2		cnvnonrel 44374	. . . . . . . . . . . . . . . 16 ⊢ ^◡(𝑋 ∖ ^◡^◡𝑋) = ∅
3		cnv0 5871	. . . . . . . . . . . . . . . 16 ⊢ ^◡∅ = ∅
4	2, 3	eqtr4i 2791	. . . . . . . . . . . . . . 15 ⊢ ^◡(𝑋 ∖ ^◡^◡𝑋) = ^◡∅
5	4	dmeqi 5896	. . . . . . . . . . . . . 14 ⊢ dom ^◡(𝑋 ∖ ^◡^◡𝑋) = dom ^◡∅
6		df-rn 5674	. . . . . . . . . . . . . 14 ⊢ ran (𝑋 ∖ ^◡^◡𝑋) = dom ^◡(𝑋 ∖ ^◡^◡𝑋)
7		df-rn 5674	. . . . . . . . . . . . . 14 ⊢ ran ∅ = dom ^◡∅
8	5, 6, 7	3eqtr4i 2798	. . . . . . . . . . . . 13 ⊢ ran (𝑋 ∖ ^◡^◡𝑋) = ran ∅
9		0ss 4357	. . . . . . . . . . . . . 14 ⊢ ∅ ⊆ ^◡𝑦
10	9	rnssi 5932	. . . . . . . . . . . . 13 ⊢ ran ∅ ⊆ ran ^◡𝑦
11	8, 10	eqsstri 3984	. . . . . . . . . . . 12 ⊢ ran (𝑋 ∖ ^◡^◡𝑋) ⊆ ran ^◡𝑦
12		ssequn2 4142	. . . . . . . . . . . 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 6144	. . . . . . . . . . 11 ⊢ ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) = (ran ^◡𝑦 ∪ ran (𝑋 ∖ ^◡^◡𝑋))
15		dfdm4 5887	. . . . . . . . . . 11 ⊢ dom 𝑦 = ran ^◡𝑦
16	13, 14, 15	3eqtr4ri 2799	. . . . . . . . . 10 ⊢ dom 𝑦 = ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))
17	4	rneqi 5929	. . . . . . . . . . . . . 14 ⊢ ran ^◡(𝑋 ∖ ^◡^◡𝑋) = ran ^◡∅
18		dfdm4 5887	. . . . . . . . . . . . . 14 ⊢ dom (𝑋 ∖ ^◡^◡𝑋) = ran ^◡(𝑋 ∖ ^◡^◡𝑋)
19		dfdm4 5887	. . . . . . . . . . . . . 14 ⊢ dom ∅ = ran ^◡∅
20	17, 18, 19	3eqtr4i 2798	. . . . . . . . . . . . 13 ⊢ dom (𝑋 ∖ ^◡^◡𝑋) = dom ∅
21		dmss 5894	. . . . . . . . . . . . . 14 ⊢ (∅ ⊆ ^◡𝑦 → dom ∅ ⊆ dom ^◡𝑦)
22	9, 21	ax-mp 5	. . . . . . . . . . . . 13 ⊢ dom ∅ ⊆ dom ^◡𝑦
23	20, 22	eqsstri 3984	. . . . . . . . . . . 12 ⊢ dom (𝑋 ∖ ^◡^◡𝑋) ⊆ dom ^◡𝑦
24		ssequn2 4142	. . . . . . . . . . . 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 5902	. . . . . . . . . . 11 ⊢ dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) = (dom ^◡𝑦 ∪ dom (𝑋 ∖ ^◡^◡𝑋))
27		df-rn 5674	. . . . . . . . . . 11 ⊢ ran 𝑦 = dom ^◡𝑦
28	25, 26, 27	3eqtr4ri 2799	. . . . . . . . . 10 ⊢ ran 𝑦 = dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))
29	16, 28	uneq12i 4120	. . . . . . . . 9 ⊢ (dom 𝑦 ∪ ran 𝑦) = (ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∪ dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))
30	29	equncomi 4114	. . . . . . . 8 ⊢ (dom 𝑦 ∪ ran 𝑦) = (dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∪ ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))
31	30	reseq2i 5977	. . . . . . 7 ⊢ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∪ ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))))
32	1, 31	eqtr2i 2789	. . . . . 6 ⊢ ( I ↾ (dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∪ ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))) = ^◡( I ↾ (dom 𝑦 ∪ ran 𝑦))
33		cnvss 5860	. . . . . 6 ⊢ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ^◡( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ ^◡𝑦)
34	32, 33	eqsstrid 3976	. . . . 5 ⊢ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∪ ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))) ⊆ ^◡𝑦)
35		ssun1 4131	. . . . 5 ⊢ ^◡𝑦 ⊆ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))
36	34, 35	sstrdi 3950	. . . 4 ⊢ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∪ ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))) ⊆ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))
37		dmeq 5895	. . . . . . 7 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → dom 𝑥 = dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))
38		rneq 5928	. . . . . . 7 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → ran 𝑥 = ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))
39	37, 38	uneq12d 4123	. . . . . 6 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → (dom 𝑥 ∪ ran 𝑥) = (dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∪ ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))))
40	39	reseq2d 5980	. . . . 5 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∪ ran (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))))
41		id 23	. . . . 5 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → 𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))
42	40, 41	sseq12d 3971	. . . 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 249	. . 3 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))
44	43	adantl 487	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))
45		cnvresid 6619	. . . . . 6 ⊢ ^◡( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑥 ∪ ran 𝑥))
46		dfdm4 5887	. . . . . . . . 9 ⊢ dom 𝑥 = ran ^◡𝑥
47		df-rn 5674	. . . . . . . . 9 ⊢ ran 𝑥 = dom ^◡𝑥
48	46, 47	uneq12i 4120	. . . . . . . 8 ⊢ (dom 𝑥 ∪ ran 𝑥) = (ran ^◡𝑥 ∪ dom ^◡𝑥)
49	48	equncomi 4114	. . . . . . 7 ⊢ (dom 𝑥 ∪ ran 𝑥) = (dom ^◡𝑥 ∪ ran ^◡𝑥)
50	49	reseq2i 5977	. . . . . 6 ⊢ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom ^◡𝑥 ∪ ran ^◡𝑥))
51	45, 50	eqtr2i 2789	. . . . 5 ⊢ ( I ↾ (dom ^◡𝑥 ∪ ran ^◡𝑥)) = ^◡( I ↾ (dom 𝑥 ∪ ran 𝑥))
52		cnvss 5860	. . . . 5 ⊢ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ^◡( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ ^◡𝑥)
53	51, 52	eqsstrid 3976	. . . 4 ⊢ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom ^◡𝑥 ∪ ran ^◡𝑥)) ⊆ ^◡𝑥)
54		dmeq 5895	. . . . . . 7 ⊢ (𝑦 = ^◡𝑥 → dom 𝑦 = dom ^◡𝑥)
55		rneq 5928	. . . . . . 7 ⊢ (𝑦 = ^◡𝑥 → ran 𝑦 = ran ^◡𝑥)
56	54, 55	uneq12d 4123	. . . . . 6 ⊢ (𝑦 = ^◡𝑥 → (dom 𝑦 ∪ ran 𝑦) = (dom ^◡𝑥 ∪ ran ^◡𝑥))
57	56	reseq2d 5980	. . . . 5 ⊢ (𝑦 = ^◡𝑥 → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom ^◡𝑥 ∪ ran ^◡𝑥)))
58		id 23	. . . . 5 ⊢ (𝑦 = ^◡𝑥 → 𝑦 = ^◡𝑥)
59	57, 58	sseq12d 3971	. . . 4 ⊢ (𝑦 = ^◡𝑥 → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ↔ ( I ↾ (dom ^◡𝑥 ∪ ran ^◡𝑥)) ⊆ ^◡𝑥))
60	53, 59	imbitrrid 249	. . 3 ⊢ (𝑦 = ^◡𝑥 → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))
61	60	adantl 487	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 = ^◡𝑥) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))
62		dmeq 5895	. . . . 5 ⊢ (𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) → dom 𝑥 = dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
63		rneq 5928	. . . . 5 ⊢ (𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) → ran 𝑥 = ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
64	62, 63	uneq12d 4123	. . . 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 5980	. . 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 23	. . 3 ⊢ (𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) → 𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
67	65, 66	sseq12d 3971	. 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 4131	. . 3 ⊢ 𝑋 ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
69	68	a1i 11	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
70		dmexg 7904	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → dom 𝑋 ∈ V)
71		rnexg 7905	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ran 𝑋 ∈ V)
72	70, 71	unexd 7759	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (dom 𝑋 ∪ ran 𝑋) ∈ V)
73	72	resiexd 7221	. . 3 ⊢ (𝑋 ∈ 𝑉 → ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ∈ V)
74		unexg 7751	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ∈ V) → (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∈ V)
75	73, 74	mpdan 700	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∈ V)
76		dmun 5902	. . . . . 6 ⊢ dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) = (dom 𝑋 ∪ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
77		ssun1 4131	. . . . . . 7 ⊢ dom 𝑋 ⊆ (dom 𝑋 ∪ ran 𝑋)
78		resdmss 6238	. . . . . . 7 ⊢ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ⊆ (dom 𝑋 ∪ ran 𝑋)
79	77, 78	unssi 4144	. . . . . 6 ⊢ (dom 𝑋 ∪ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
80	76, 79	eqsstri 3984	. . . . 5 ⊢ dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
81		rnun 6144	. . . . . 6 ⊢ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) = (ran 𝑋 ∪ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
82		ssun2 4132	. . . . . . 7 ⊢ ran 𝑋 ⊆ (dom 𝑋 ∪ ran 𝑋)
83		rnresi 6079	. . . . . . . 8 ⊢ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)) = (dom 𝑋 ∪ ran 𝑋)
84	83	eqimssi 3998	. . . . . . 7 ⊢ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ⊆ (dom 𝑋 ∪ ran 𝑋)
85	82, 84	unssi 4144	. . . . . 6 ⊢ (ran 𝑋 ∪ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
86	81, 85	eqsstri 3984	. . . . 5 ⊢ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
87	80, 86	pm3.2i 476	. . . 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 4143	. . . . 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 6005	. . . . 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 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)))
91		ssun4 4134	. . . 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 44409	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 {cab 2743 Vcvv 3457 ∖ cdif 3903 ∪ cun 3904 ⊆ wss 3906 ∅c0 4286 ∩ cint 4914 I cid 5557 ^◡ccnv 5662 dom cdm 5663 ran crn 5664 ↾ cres 5665

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-1st 7992 df-2nd 7993

This theorem is used by: (None)

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