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Theorem cnvrcl0 37399
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 5928 . . . . . . 7 ( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom 𝑦 ∪ ran 𝑦))
2 cnvnonrel 37361 . . . . . . . . . . . . . . . 16 (𝑋𝑋) = ∅
3 cnv0 5498 . . . . . . . . . . . . . . . 16 ∅ = ∅
42, 3eqtr4i 2651 . . . . . . . . . . . . . . 15 (𝑋𝑋) =
54dmeqi 5290 . . . . . . . . . . . . . 14 dom (𝑋𝑋) = dom
6 df-rn 5090 . . . . . . . . . . . . . 14 ran (𝑋𝑋) = dom (𝑋𝑋)
7 df-rn 5090 . . . . . . . . . . . . . 14 ran ∅ = dom
85, 6, 73eqtr4i 2658 . . . . . . . . . . . . 13 ran (𝑋𝑋) = ran ∅
9 0ss 3949 . . . . . . . . . . . . . 14 ∅ ⊆ 𝑦
10 rnss 5318 . . . . . . . . . . . . . 14 (∅ ⊆ 𝑦 → ran ∅ ⊆ ran 𝑦)
119, 10ax-mp 5 . . . . . . . . . . . . 13 ran ∅ ⊆ ran 𝑦
128, 11eqsstri 3619 . . . . . . . . . . . 12 ran (𝑋𝑋) ⊆ ran 𝑦
13 ssequn2 3769 . . . . . . . . . . . 12 (ran (𝑋𝑋) ⊆ ran 𝑦 ↔ (ran 𝑦 ∪ ran (𝑋𝑋)) = ran 𝑦)
1412, 13mpbi 220 . . . . . . . . . . 11 (ran 𝑦 ∪ ran (𝑋𝑋)) = ran 𝑦
15 rnun 5504 . . . . . . . . . . 11 ran (𝑦 ∪ (𝑋𝑋)) = (ran 𝑦 ∪ ran (𝑋𝑋))
16 dfdm4 5281 . . . . . . . . . . 11 dom 𝑦 = ran 𝑦
1714, 15, 163eqtr4ri 2659 . . . . . . . . . 10 dom 𝑦 = ran (𝑦 ∪ (𝑋𝑋))
184rneqi 5316 . . . . . . . . . . . . . 14 ran (𝑋𝑋) = ran
19 dfdm4 5281 . . . . . . . . . . . . . 14 dom (𝑋𝑋) = ran (𝑋𝑋)
20 dfdm4 5281 . . . . . . . . . . . . . 14 dom ∅ = ran
2118, 19, 203eqtr4i 2658 . . . . . . . . . . . . 13 dom (𝑋𝑋) = dom ∅
22 dmss 5288 . . . . . . . . . . . . . 14 (∅ ⊆ 𝑦 → dom ∅ ⊆ dom 𝑦)
239, 22ax-mp 5 . . . . . . . . . . . . 13 dom ∅ ⊆ dom 𝑦
2421, 23eqsstri 3619 . . . . . . . . . . . 12 dom (𝑋𝑋) ⊆ dom 𝑦
25 ssequn2 3769 . . . . . . . . . . . 12 (dom (𝑋𝑋) ⊆ dom 𝑦 ↔ (dom 𝑦 ∪ dom (𝑋𝑋)) = dom 𝑦)
2624, 25mpbi 220 . . . . . . . . . . 11 (dom 𝑦 ∪ dom (𝑋𝑋)) = dom 𝑦
27 dmun 5296 . . . . . . . . . . 11 dom (𝑦 ∪ (𝑋𝑋)) = (dom 𝑦 ∪ dom (𝑋𝑋))
28 df-rn 5090 . . . . . . . . . . 11 ran 𝑦 = dom 𝑦
2926, 27, 283eqtr4ri 2659 . . . . . . . . . 10 ran 𝑦 = dom (𝑦 ∪ (𝑋𝑋))
3017, 29uneq12i 3748 . . . . . . . . 9 (dom 𝑦 ∪ ran 𝑦) = (ran (𝑦 ∪ (𝑋𝑋)) ∪ dom (𝑦 ∪ (𝑋𝑋)))
3130equncomi 3742 . . . . . . . 8 (dom 𝑦 ∪ ran 𝑦) = (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋)))
3231reseq2i 5357 . . . . . . 7 ( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋))))
331, 32eqtr2i 2649 . . . . . 6 ( I ↾ (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋)))) = ( I ↾ (dom 𝑦 ∪ ran 𝑦))
34 cnvss 5259 . . . . . 6 (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)
3533, 34syl5eqss 3633 . . . . 5 (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋)))) ⊆ 𝑦)
36 ssun1 3759 . . . . 5 𝑦 ⊆ (𝑦 ∪ (𝑋𝑋))
3735, 36syl6ss 3600 . . . 4 (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋)))) ⊆ (𝑦 ∪ (𝑋𝑋)))
38 dmeq 5289 . . . . . . 7 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → dom 𝑥 = dom (𝑦 ∪ (𝑋𝑋)))
39 rneq 5315 . . . . . . 7 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → ran 𝑥 = ran (𝑦 ∪ (𝑋𝑋)))
4038, 39uneq12d 3751 . . . . . 6 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → (dom 𝑥 ∪ ran 𝑥) = (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋))))
4140reseq2d 5360 . . . . 5 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋)))))
42 id 22 . . . . 5 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → 𝑥 = (𝑦 ∪ (𝑋𝑋)))
4341, 42sseq12d 3618 . . . 4 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 ↔ ( I ↾ (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋)))) ⊆ (𝑦 ∪ (𝑋𝑋))))
4437, 43syl5ibr 236 . . 3 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))
4544adantl 482 . 2 ((𝑋𝑉𝑥 = (𝑦 ∪ (𝑋𝑋))) → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))
46 cnvresid 5928 . . . . . 6 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑥 ∪ ran 𝑥))
47 dfdm4 5281 . . . . . . . . 9 dom 𝑥 = ran 𝑥
48 df-rn 5090 . . . . . . . . 9 ran 𝑥 = dom 𝑥
4947, 48uneq12i 3748 . . . . . . . 8 (dom 𝑥 ∪ ran 𝑥) = (ran 𝑥 ∪ dom 𝑥)
5049equncomi 3742 . . . . . . 7 (dom 𝑥 ∪ ran 𝑥) = (dom 𝑥 ∪ ran 𝑥)
5150reseq2i 5357 . . . . . 6 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑥 ∪ ran 𝑥))
5246, 51eqtr2i 2649 . . . . 5 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑥 ∪ ran 𝑥))
53 cnvss 5259 . . . . 5 (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)
5452, 53syl5eqss 3633 . . . 4 (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)
55 dmeq 5289 . . . . . . 7 (𝑦 = 𝑥 → dom 𝑦 = dom 𝑥)
56 rneq 5315 . . . . . . 7 (𝑦 = 𝑥 → ran 𝑦 = ran 𝑥)
5755, 56uneq12d 3751 . . . . . 6 (𝑦 = 𝑥 → (dom 𝑦 ∪ ran 𝑦) = (dom 𝑥 ∪ ran 𝑥))
5857reseq2d 5360 . . . . 5 (𝑦 = 𝑥 → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom 𝑥 ∪ ran 𝑥)))
59 id 22 . . . . 5 (𝑦 = 𝑥𝑦 = 𝑥)
6058, 59sseq12d 3618 . . . 4 (𝑦 = 𝑥 → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ↔ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))
6154, 60syl5ibr 236 . . 3 (𝑦 = 𝑥 → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))
6261adantl 482 . 2 ((𝑋𝑉𝑦 = 𝑥) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))
63 dmeq 5289 . . . . 5 (𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) → dom 𝑥 = dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
64 rneq 5315 . . . . 5 (𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) → ran 𝑥 = ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
6563, 64uneq12d 3751 . . . 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)))))
6665reseq2d 5360 . . 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))))
6866, 67sseq12d 3618 . 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 3759 . . 3 𝑋 ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
7069a1i 11 . 2 (𝑋𝑉𝑋 ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
71 dmexg 7045 . . . . 5 (𝑋𝑉 → dom 𝑋 ∈ V)
72 rnexg 7046 . . . . 5 (𝑋𝑉 → ran 𝑋 ∈ V)
73 unexg 6913 . . . . 5 ((dom 𝑋 ∈ V ∧ ran 𝑋 ∈ V) → (dom 𝑋 ∪ ran 𝑋) ∈ V)
7471, 72, 73syl2anc 692 . . . 4 (𝑋𝑉 → (dom 𝑋 ∪ ran 𝑋) ∈ V)
7574resiexd 6435 . . 3 (𝑋𝑉 → ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ∈ V)
76 unexg 6913 . . 3 ((𝑋𝑉 ∧ ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ∈ V) → (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∈ V)
7775, 76mpdan 701 . 2 (𝑋𝑉 → (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∈ V)
78 dmun 5296 . . . . . 6 dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) = (dom 𝑋 ∪ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
79 ssun1 3759 . . . . . . 7 dom 𝑋 ⊆ (dom 𝑋 ∪ ran 𝑋)
80 dmresi 5420 . . . . . . . 8 dom ( I ↾ (dom 𝑋 ∪ ran 𝑋)) = (dom 𝑋 ∪ ran 𝑋)
8180eqimssi 3643 . . . . . . 7 dom ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ⊆ (dom 𝑋 ∪ ran 𝑋)
8279, 81unssi 3771 . . . . . 6 (dom 𝑋 ∪ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
8378, 82eqsstri 3619 . . . . 5 dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
84 rnun 5504 . . . . . 6 ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) = (ran 𝑋 ∪ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
85 ssun2 3760 . . . . . . 7 ran 𝑋 ⊆ (dom 𝑋 ∪ ran 𝑋)
86 rnresi 5442 . . . . . . . 8 ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)) = (dom 𝑋 ∪ ran 𝑋)
8786eqimssi 3643 . . . . . . 7 ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ⊆ (dom 𝑋 ∪ ran 𝑋)
8885, 87unssi 3771 . . . . . 6 (ran 𝑋 ∪ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
8984, 88eqsstri 3619 . . . . 5 ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
9083, 89pm3.2i 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 3770 . . . . 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 5388 . . . . 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)))
9391, 92sylbi 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 3762 . . . 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))))
9590, 93, 94mp2b 10 . . 3 ( I ↾ (dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∪ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))) ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
9695a1i 11 . 2 (𝑋𝑉 → ( I ↾ (dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∪ ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))) ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
9745, 62, 68, 70, 77, 96clcnvlem 37397 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 384   = wceq 1480  wcel 1992  {cab 2612  Vcvv 3191  cdif 3557  cun 3558  wss 3560  c0 3896   cint 4445   I cid 4989  ccnv 5078  dom cdm 5079  ran crn 5080  cres 5081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-1st 7116  df-2nd 7117
This theorem is referenced by: (None)
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