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Theorem cnvrcl0 42376
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 (𝑋𝑉 {𝑥 ∣ (𝑋𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} = {𝑦 ∣ (𝑋𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)})
Distinct variable groups:   𝑥,𝑦,𝑉   𝑥,𝑋,𝑦

Proof of Theorem cnvrcl0
StepHypRef Expression
1 cnvresid 6628 . . . . . . 7 ( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom 𝑦 ∪ ran 𝑦))
2 cnvnonrel 42339 . . . . . . . . . . . . . . . 16 (𝑋𝑋) = ∅
3 cnv0 6141 . . . . . . . . . . . . . . . 16 ∅ = ∅
42, 3eqtr4i 2764 . . . . . . . . . . . . . . 15 (𝑋𝑋) =
54dmeqi 5905 . . . . . . . . . . . . . 14 dom (𝑋𝑋) = dom
6 df-rn 5688 . . . . . . . . . . . . . 14 ran (𝑋𝑋) = dom (𝑋𝑋)
7 df-rn 5688 . . . . . . . . . . . . . 14 ran ∅ = dom
85, 6, 73eqtr4i 2771 . . . . . . . . . . . . 13 ran (𝑋𝑋) = ran ∅
9 0ss 4397 . . . . . . . . . . . . . 14 ∅ ⊆ 𝑦
109rnssi 5940 . . . . . . . . . . . . 13 ran ∅ ⊆ ran 𝑦
118, 10eqsstri 4017 . . . . . . . . . . . 12 ran (𝑋𝑋) ⊆ ran 𝑦
12 ssequn2 4184 . . . . . . . . . . . 12 (ran (𝑋𝑋) ⊆ ran 𝑦 ↔ (ran 𝑦 ∪ ran (𝑋𝑋)) = ran 𝑦)
1311, 12mpbi 229 . . . . . . . . . . 11 (ran 𝑦 ∪ ran (𝑋𝑋)) = ran 𝑦
14 rnun 6146 . . . . . . . . . . 11 ran (𝑦 ∪ (𝑋𝑋)) = (ran 𝑦 ∪ ran (𝑋𝑋))
15 dfdm4 5896 . . . . . . . . . . 11 dom 𝑦 = ran 𝑦
1613, 14, 153eqtr4ri 2772 . . . . . . . . . 10 dom 𝑦 = ran (𝑦 ∪ (𝑋𝑋))
174rneqi 5937 . . . . . . . . . . . . . 14 ran (𝑋𝑋) = ran
18 dfdm4 5896 . . . . . . . . . . . . . 14 dom (𝑋𝑋) = ran (𝑋𝑋)
19 dfdm4 5896 . . . . . . . . . . . . . 14 dom ∅ = ran
2017, 18, 193eqtr4i 2771 . . . . . . . . . . . . 13 dom (𝑋𝑋) = dom ∅
21 dmss 5903 . . . . . . . . . . . . . 14 (∅ ⊆ 𝑦 → dom ∅ ⊆ dom 𝑦)
229, 21ax-mp 5 . . . . . . . . . . . . 13 dom ∅ ⊆ dom 𝑦
2320, 22eqsstri 4017 . . . . . . . . . . . 12 dom (𝑋𝑋) ⊆ dom 𝑦
24 ssequn2 4184 . . . . . . . . . . . 12 (dom (𝑋𝑋) ⊆ dom 𝑦 ↔ (dom 𝑦 ∪ dom (𝑋𝑋)) = dom 𝑦)
2523, 24mpbi 229 . . . . . . . . . . 11 (dom 𝑦 ∪ dom (𝑋𝑋)) = dom 𝑦
26 dmun 5911 . . . . . . . . . . 11 dom (𝑦 ∪ (𝑋𝑋)) = (dom 𝑦 ∪ dom (𝑋𝑋))
27 df-rn 5688 . . . . . . . . . . 11 ran 𝑦 = dom 𝑦
2825, 26, 273eqtr4ri 2772 . . . . . . . . . 10 ran 𝑦 = dom (𝑦 ∪ (𝑋𝑋))
2916, 28uneq12i 4162 . . . . . . . . 9 (dom 𝑦 ∪ ran 𝑦) = (ran (𝑦 ∪ (𝑋𝑋)) ∪ dom (𝑦 ∪ (𝑋𝑋)))
3029equncomi 4156 . . . . . . . 8 (dom 𝑦 ∪ ran 𝑦) = (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋)))
3130reseq2i 5979 . . . . . . 7 ( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋))))
321, 31eqtr2i 2762 . . . . . 6 ( I ↾ (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋)))) = ( I ↾ (dom 𝑦 ∪ ran 𝑦))
33 cnvss 5873 . . . . . 6 (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)
3432, 33eqsstrid 4031 . . . . 5 (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋)))) ⊆ 𝑦)
35 ssun1 4173 . . . . 5 𝑦 ⊆ (𝑦 ∪ (𝑋𝑋))
3634, 35sstrdi 3995 . . . 4 (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋)))) ⊆ (𝑦 ∪ (𝑋𝑋)))
37 dmeq 5904 . . . . . . 7 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → dom 𝑥 = dom (𝑦 ∪ (𝑋𝑋)))
38 rneq 5936 . . . . . . 7 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → ran 𝑥 = ran (𝑦 ∪ (𝑋𝑋)))
3937, 38uneq12d 4165 . . . . . 6 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → (dom 𝑥 ∪ ran 𝑥) = (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋))))
4039reseq2d 5982 . . . . 5 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋)))))
41 id 22 . . . . 5 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → 𝑥 = (𝑦 ∪ (𝑋𝑋)))
4240, 41sseq12d 4016 . . . 4 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 ↔ ( I ↾ (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋)))) ⊆ (𝑦 ∪ (𝑋𝑋))))
4336, 42imbitrrid 245 . . 3 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))
4443adantl 483 . 2 ((𝑋𝑉𝑥 = (𝑦 ∪ (𝑋𝑋))) → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))
45 cnvresid 6628 . . . . . 6 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑥 ∪ ran 𝑥))
46 dfdm4 5896 . . . . . . . . 9 dom 𝑥 = ran 𝑥
47 df-rn 5688 . . . . . . . . 9 ran 𝑥 = dom 𝑥
4846, 47uneq12i 4162 . . . . . . . 8 (dom 𝑥 ∪ ran 𝑥) = (ran 𝑥 ∪ dom 𝑥)
4948equncomi 4156 . . . . . . 7 (dom 𝑥 ∪ ran 𝑥) = (dom 𝑥 ∪ ran 𝑥)
5049reseq2i 5979 . . . . . 6 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑥 ∪ ran 𝑥))
5145, 50eqtr2i 2762 . . . . 5 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑥 ∪ ran 𝑥))
52 cnvss 5873 . . . . 5 (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)
5351, 52eqsstrid 4031 . . . 4 (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)
54 dmeq 5904 . . . . . . 7 (𝑦 = 𝑥 → dom 𝑦 = dom 𝑥)
55 rneq 5936 . . . . . . 7 (𝑦 = 𝑥 → ran 𝑦 = ran 𝑥)
5654, 55uneq12d 4165 . . . . . 6 (𝑦 = 𝑥 → (dom 𝑦 ∪ ran 𝑦) = (dom 𝑥 ∪ ran 𝑥))
5756reseq2d 5982 . . . . 5 (𝑦 = 𝑥 → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom 𝑥 ∪ ran 𝑥)))
58 id 22 . . . . 5 (𝑦 = 𝑥𝑦 = 𝑥)
5957, 58sseq12d 4016 . . . 4 (𝑦 = 𝑥 → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ↔ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))
6053, 59imbitrrid 245 . . 3 (𝑦 = 𝑥 → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))
6160adantl 483 . 2 ((𝑋𝑉𝑦 = 𝑥) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))
62 dmeq 5904 . . . . 5 (𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) → dom 𝑥 = dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
63 rneq 5936 . . . . 5 (𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) → ran 𝑥 = ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
6462, 63uneq12d 4165 . . . 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)))))
6564reseq2d 5982 . . 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))))
6765, 66sseq12d 4016 . 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 4173 . . 3 𝑋 ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
6968a1i 11 . 2 (𝑋𝑉𝑋 ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
70 dmexg 7894 . . . . 5 (𝑋𝑉 → dom 𝑋 ∈ V)
71 rnexg 7895 . . . . 5 (𝑋𝑉 → ran 𝑋 ∈ V)
7270, 71unexd 7741 . . . 4 (𝑋𝑉 → (dom 𝑋 ∪ ran 𝑋) ∈ V)
7372resiexd 7218 . . 3 (𝑋𝑉 → ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ∈ V)
74 unexg 7736 . . 3 ((𝑋𝑉 ∧ ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ∈ V) → (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∈ V)
7573, 74mpdan 686 . 2 (𝑋𝑉 → (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∈ V)
76 dmun 5911 . . . . . 6 dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) = (dom 𝑋 ∪ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
77 ssun1 4173 . . . . . . 7 dom 𝑋 ⊆ (dom 𝑋 ∪ ran 𝑋)
78 resdmss 6235 . . . . . . 7 dom ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ⊆ (dom 𝑋 ∪ ran 𝑋)
7977, 78unssi 4186 . . . . . 6 (dom 𝑋 ∪ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
8076, 79eqsstri 4017 . . . . 5 dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
81 rnun 6146 . . . . . 6 ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) = (ran 𝑋 ∪ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
82 ssun2 4174 . . . . . . 7 ran 𝑋 ⊆ (dom 𝑋 ∪ ran 𝑋)
83 rnresi 6075 . . . . . . . 8 ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)) = (dom 𝑋 ∪ ran 𝑋)
8483eqimssi 4043 . . . . . . 7 ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ⊆ (dom 𝑋 ∪ ran 𝑋)
8582, 84unssi 4186 . . . . . 6 (ran 𝑋 ∪ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
8681, 85eqsstri 4017 . . . . 5 ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
8780, 86pm3.2i 472 . . . 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 4185 . . . . 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 6010 . . . . 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)))
9088, 89sylbi 216 . . . 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 4176 . . . 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))))
9287, 90, 91mp2b 10 . . 3 ( I ↾ (dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∪ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))) ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
9392a1i 11 . 2 (𝑋𝑉 → ( I ↾ (dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∪ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))) ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
9444, 61, 67, 69, 75, 93clcnvlem 42374 1 (𝑋𝑉 {𝑥 ∣ (𝑋𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} = {𝑦 ∣ (𝑋𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {cab 2710  Vcvv 3475  cdif 3946  cun 3947  wss 3949  c0 4323   cint 4951   I cid 5574  ccnv 5676  dom cdm 5677  ran crn 5678  cres 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-1st 7975  df-2nd 7976
This theorem is referenced by: (None)
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