Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cnvrcl0 Structured version   Visualization version   GIF version

Theorem cnvrcl0 40720
 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 6414 . . . . . . 7 ( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom 𝑦 ∪ ran 𝑦))
2 cnvnonrel 40683 . . . . . . . . . . . . . . . 16 (𝑋𝑋) = ∅
3 cnv0 5971 . . . . . . . . . . . . . . . 16 ∅ = ∅
42, 3eqtr4i 2784 . . . . . . . . . . . . . . 15 (𝑋𝑋) =
54dmeqi 5744 . . . . . . . . . . . . . 14 dom (𝑋𝑋) = dom
6 df-rn 5535 . . . . . . . . . . . . . 14 ran (𝑋𝑋) = dom (𝑋𝑋)
7 df-rn 5535 . . . . . . . . . . . . . 14 ran ∅ = dom
85, 6, 73eqtr4i 2791 . . . . . . . . . . . . 13 ran (𝑋𝑋) = ran ∅
9 0ss 4292 . . . . . . . . . . . . . 14 ∅ ⊆ 𝑦
109rnssi 5781 . . . . . . . . . . . . 13 ran ∅ ⊆ ran 𝑦
118, 10eqsstri 3926 . . . . . . . . . . . 12 ran (𝑋𝑋) ⊆ ran 𝑦
12 ssequn2 4088 . . . . . . . . . . . 12 (ran (𝑋𝑋) ⊆ ran 𝑦 ↔ (ran 𝑦 ∪ ran (𝑋𝑋)) = ran 𝑦)
1311, 12mpbi 233 . . . . . . . . . . 11 (ran 𝑦 ∪ ran (𝑋𝑋)) = ran 𝑦
14 rnun 5976 . . . . . . . . . . 11 ran (𝑦 ∪ (𝑋𝑋)) = (ran 𝑦 ∪ ran (𝑋𝑋))
15 dfdm4 5735 . . . . . . . . . . 11 dom 𝑦 = ran 𝑦
1613, 14, 153eqtr4ri 2792 . . . . . . . . . 10 dom 𝑦 = ran (𝑦 ∪ (𝑋𝑋))
174rneqi 5778 . . . . . . . . . . . . . 14 ran (𝑋𝑋) = ran
18 dfdm4 5735 . . . . . . . . . . . . . 14 dom (𝑋𝑋) = ran (𝑋𝑋)
19 dfdm4 5735 . . . . . . . . . . . . . 14 dom ∅ = ran
2017, 18, 193eqtr4i 2791 . . . . . . . . . . . . 13 dom (𝑋𝑋) = dom ∅
21 dmss 5742 . . . . . . . . . . . . . 14 (∅ ⊆ 𝑦 → dom ∅ ⊆ dom 𝑦)
229, 21ax-mp 5 . . . . . . . . . . . . 13 dom ∅ ⊆ dom 𝑦
2320, 22eqsstri 3926 . . . . . . . . . . . 12 dom (𝑋𝑋) ⊆ dom 𝑦
24 ssequn2 4088 . . . . . . . . . . . 12 (dom (𝑋𝑋) ⊆ dom 𝑦 ↔ (dom 𝑦 ∪ dom (𝑋𝑋)) = dom 𝑦)
2523, 24mpbi 233 . . . . . . . . . . 11 (dom 𝑦 ∪ dom (𝑋𝑋)) = dom 𝑦
26 dmun 5750 . . . . . . . . . . 11 dom (𝑦 ∪ (𝑋𝑋)) = (dom 𝑦 ∪ dom (𝑋𝑋))
27 df-rn 5535 . . . . . . . . . . 11 ran 𝑦 = dom 𝑦
2825, 26, 273eqtr4ri 2792 . . . . . . . . . 10 ran 𝑦 = dom (𝑦 ∪ (𝑋𝑋))
2916, 28uneq12i 4066 . . . . . . . . 9 (dom 𝑦 ∪ ran 𝑦) = (ran (𝑦 ∪ (𝑋𝑋)) ∪ dom (𝑦 ∪ (𝑋𝑋)))
3029equncomi 4060 . . . . . . . 8 (dom 𝑦 ∪ ran 𝑦) = (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋)))
3130reseq2i 5820 . . . . . . 7 ( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋))))
321, 31eqtr2i 2782 . . . . . 6 ( I ↾ (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋)))) = ( I ↾ (dom 𝑦 ∪ ran 𝑦))
33 cnvss 5712 . . . . . 6 (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)
3432, 33eqsstrid 3940 . . . . 5 (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋)))) ⊆ 𝑦)
35 ssun1 4077 . . . . 5 𝑦 ⊆ (𝑦 ∪ (𝑋𝑋))
3634, 35sstrdi 3904 . . . 4 (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋)))) ⊆ (𝑦 ∪ (𝑋𝑋)))
37 dmeq 5743 . . . . . . 7 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → dom 𝑥 = dom (𝑦 ∪ (𝑋𝑋)))
38 rneq 5777 . . . . . . 7 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → ran 𝑥 = ran (𝑦 ∪ (𝑋𝑋)))
3937, 38uneq12d 4069 . . . . . 6 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → (dom 𝑥 ∪ ran 𝑥) = (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋))))
4039reseq2d 5823 . . . . 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 3925 . . . 4 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 ↔ ( I ↾ (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋)))) ⊆ (𝑦 ∪ (𝑋𝑋))))
4336, 42syl5ibr 249 . . 3 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))
4443adantl 485 . 2 ((𝑋𝑉𝑥 = (𝑦 ∪ (𝑋𝑋))) → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))
45 cnvresid 6414 . . . . . 6 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑥 ∪ ran 𝑥))
46 dfdm4 5735 . . . . . . . . 9 dom 𝑥 = ran 𝑥
47 df-rn 5535 . . . . . . . . 9 ran 𝑥 = dom 𝑥
4846, 47uneq12i 4066 . . . . . . . 8 (dom 𝑥 ∪ ran 𝑥) = (ran 𝑥 ∪ dom 𝑥)
4948equncomi 4060 . . . . . . 7 (dom 𝑥 ∪ ran 𝑥) = (dom 𝑥 ∪ ran 𝑥)
5049reseq2i 5820 . . . . . 6 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑥 ∪ ran 𝑥))
5145, 50eqtr2i 2782 . . . . 5 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑥 ∪ ran 𝑥))
52 cnvss 5712 . . . . 5 (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)
5351, 52eqsstrid 3940 . . . 4 (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)
54 dmeq 5743 . . . . . . 7 (𝑦 = 𝑥 → dom 𝑦 = dom 𝑥)
55 rneq 5777 . . . . . . 7 (𝑦 = 𝑥 → ran 𝑦 = ran 𝑥)
5654, 55uneq12d 4069 . . . . . 6 (𝑦 = 𝑥 → (dom 𝑦 ∪ ran 𝑦) = (dom 𝑥 ∪ ran 𝑥))
5756reseq2d 5823 . . . . 5 (𝑦 = 𝑥 → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom 𝑥 ∪ ran 𝑥)))
58 id 22 . . . . 5 (𝑦 = 𝑥𝑦 = 𝑥)
5957, 58sseq12d 3925 . . . 4 (𝑦 = 𝑥 → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ↔ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))
6053, 59syl5ibr 249 . . 3 (𝑦 = 𝑥 → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))
6160adantl 485 . 2 ((𝑋𝑉𝑦 = 𝑥) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))
62 dmeq 5743 . . . . 5 (𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) → dom 𝑥 = dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
63 rneq 5777 . . . . 5 (𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) → ran 𝑥 = ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
6462, 63uneq12d 4069 . . . 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 5823 . . 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 3925 . 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 4077 . . 3 𝑋 ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
6968a1i 11 . 2 (𝑋𝑉𝑋 ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
70 dmexg 7613 . . . . 5 (𝑋𝑉 → dom 𝑋 ∈ V)
71 rnexg 7614 . . . . 5 (𝑋𝑉 → ran 𝑋 ∈ V)
72 unexg 7470 . . . . 5 ((dom 𝑋 ∈ V ∧ ran 𝑋 ∈ V) → (dom 𝑋 ∪ ran 𝑋) ∈ V)
7370, 71, 72syl2anc 587 . . . 4 (𝑋𝑉 → (dom 𝑋 ∪ ran 𝑋) ∈ V)
7473resiexd 6970 . . 3 (𝑋𝑉 → ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ∈ V)
75 unexg 7470 . . 3 ((𝑋𝑉 ∧ ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ∈ V) → (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∈ V)
7674, 75mpdan 686 . 2 (𝑋𝑉 → (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∈ V)
77 dmun 5750 . . . . . 6 dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) = (dom 𝑋 ∪ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
78 ssun1 4077 . . . . . . 7 dom 𝑋 ⊆ (dom 𝑋 ∪ ran 𝑋)
79 dmresi 5893 . . . . . . . 8 dom ( I ↾ (dom 𝑋 ∪ ran 𝑋)) = (dom 𝑋 ∪ ran 𝑋)
8079eqimssi 3950 . . . . . . 7 dom ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ⊆ (dom 𝑋 ∪ ran 𝑋)
8178, 80unssi 4090 . . . . . 6 (dom 𝑋 ∪ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
8277, 81eqsstri 3926 . . . . 5 dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
83 rnun 5976 . . . . . 6 ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) = (ran 𝑋 ∪ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
84 ssun2 4078 . . . . . . 7 ran 𝑋 ⊆ (dom 𝑋 ∪ ran 𝑋)
85 rnresi 5915 . . . . . . . 8 ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)) = (dom 𝑋 ∪ ran 𝑋)
8685eqimssi 3950 . . . . . . 7 ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ⊆ (dom 𝑋 ∪ ran 𝑋)
8784, 86unssi 4090 . . . . . 6 (ran 𝑋 ∪ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
8883, 87eqsstri 3926 . . . . 5 ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
8982, 88pm3.2i 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 4089 . . . . 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 5851 . . . . 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)))
9290, 91sylbi 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 4080 . . . 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))))
9489, 92, 93mp2b 10 . . 3 ( I ↾ (dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∪ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))) ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
9594a1i 11 . 2 (𝑋𝑉 → ( I ↾ (dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∪ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))) ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
9644, 61, 67, 69, 76, 95clcnvlem 40718 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 399   = wceq 1538   ∈ wcel 2111  {cab 2735  Vcvv 3409   ∖ cdif 3855   ∪ cun 3856   ⊆ wss 3858  ∅c0 4225  ∩ cint 4838   I cid 5429  ◡ccnv 5523  dom cdm 5524  ran crn 5525   ↾ cres 5526 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 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-1st 7693  df-2nd 7694 This theorem is referenced by: (None)
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