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Theorem cnvrcl0 38249
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 6006 . . . . . . 7 ( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom 𝑦 ∪ ran 𝑦))
2 cnvnonrel 38211 . . . . . . . . . . . . . . . 16 (𝑋𝑋) = ∅
3 cnv0 5570 . . . . . . . . . . . . . . . 16 ∅ = ∅
42, 3eqtr4i 2676 . . . . . . . . . . . . . . 15 (𝑋𝑋) =
54dmeqi 5357 . . . . . . . . . . . . . 14 dom (𝑋𝑋) = dom
6 df-rn 5154 . . . . . . . . . . . . . 14 ran (𝑋𝑋) = dom (𝑋𝑋)
7 df-rn 5154 . . . . . . . . . . . . . 14 ran ∅ = dom
85, 6, 73eqtr4i 2683 . . . . . . . . . . . . 13 ran (𝑋𝑋) = ran ∅
9 0ss 4005 . . . . . . . . . . . . . 14 ∅ ⊆ 𝑦
10 rnss 5386 . . . . . . . . . . . . . 14 (∅ ⊆ 𝑦 → ran ∅ ⊆ ran 𝑦)
119, 10ax-mp 5 . . . . . . . . . . . . 13 ran ∅ ⊆ ran 𝑦
128, 11eqsstri 3668 . . . . . . . . . . . 12 ran (𝑋𝑋) ⊆ ran 𝑦
13 ssequn2 3819 . . . . . . . . . . . 12 (ran (𝑋𝑋) ⊆ ran 𝑦 ↔ (ran 𝑦 ∪ ran (𝑋𝑋)) = ran 𝑦)
1412, 13mpbi 220 . . . . . . . . . . 11 (ran 𝑦 ∪ ran (𝑋𝑋)) = ran 𝑦
15 rnun 5576 . . . . . . . . . . 11 ran (𝑦 ∪ (𝑋𝑋)) = (ran 𝑦 ∪ ran (𝑋𝑋))
16 dfdm4 5348 . . . . . . . . . . 11 dom 𝑦 = ran 𝑦
1714, 15, 163eqtr4ri 2684 . . . . . . . . . 10 dom 𝑦 = ran (𝑦 ∪ (𝑋𝑋))
184rneqi 5384 . . . . . . . . . . . . . 14 ran (𝑋𝑋) = ran
19 dfdm4 5348 . . . . . . . . . . . . . 14 dom (𝑋𝑋) = ran (𝑋𝑋)
20 dfdm4 5348 . . . . . . . . . . . . . 14 dom ∅ = ran
2118, 19, 203eqtr4i 2683 . . . . . . . . . . . . 13 dom (𝑋𝑋) = dom ∅
22 dmss 5355 . . . . . . . . . . . . . 14 (∅ ⊆ 𝑦 → dom ∅ ⊆ dom 𝑦)
239, 22ax-mp 5 . . . . . . . . . . . . 13 dom ∅ ⊆ dom 𝑦
2421, 23eqsstri 3668 . . . . . . . . . . . 12 dom (𝑋𝑋) ⊆ dom 𝑦
25 ssequn2 3819 . . . . . . . . . . . 12 (dom (𝑋𝑋) ⊆ dom 𝑦 ↔ (dom 𝑦 ∪ dom (𝑋𝑋)) = dom 𝑦)
2624, 25mpbi 220 . . . . . . . . . . 11 (dom 𝑦 ∪ dom (𝑋𝑋)) = dom 𝑦
27 dmun 5363 . . . . . . . . . . 11 dom (𝑦 ∪ (𝑋𝑋)) = (dom 𝑦 ∪ dom (𝑋𝑋))
28 df-rn 5154 . . . . . . . . . . 11 ran 𝑦 = dom 𝑦
2926, 27, 283eqtr4ri 2684 . . . . . . . . . 10 ran 𝑦 = dom (𝑦 ∪ (𝑋𝑋))
3017, 29uneq12i 3798 . . . . . . . . 9 (dom 𝑦 ∪ ran 𝑦) = (ran (𝑦 ∪ (𝑋𝑋)) ∪ dom (𝑦 ∪ (𝑋𝑋)))
3130equncomi 3792 . . . . . . . 8 (dom 𝑦 ∪ ran 𝑦) = (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋)))
3231reseq2i 5425 . . . . . . 7 ( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋))))
331, 32eqtr2i 2674 . . . . . 6 ( I ↾ (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋)))) = ( I ↾ (dom 𝑦 ∪ ran 𝑦))
34 cnvss 5327 . . . . . 6 (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)
3533, 34syl5eqss 3682 . . . . 5 (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋)))) ⊆ 𝑦)
36 ssun1 3809 . . . . 5 𝑦 ⊆ (𝑦 ∪ (𝑋𝑋))
3735, 36syl6ss 3648 . . . 4 (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋)))) ⊆ (𝑦 ∪ (𝑋𝑋)))
38 dmeq 5356 . . . . . . 7 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → dom 𝑥 = dom (𝑦 ∪ (𝑋𝑋)))
39 rneq 5383 . . . . . . 7 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → ran 𝑥 = ran (𝑦 ∪ (𝑋𝑋)))
4038, 39uneq12d 3801 . . . . . 6 (𝑥 = (𝑦 ∪ (𝑋𝑋)) → (dom 𝑥 ∪ ran 𝑥) = (dom (𝑦 ∪ (𝑋𝑋)) ∪ ran (𝑦 ∪ (𝑋𝑋))))
4140reseq2d 5428 . . . . 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 3667 . . . 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 481 . 2 ((𝑋𝑉𝑥 = (𝑦 ∪ (𝑋𝑋))) → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))
46 cnvresid 6006 . . . . . 6 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑥 ∪ ran 𝑥))
47 dfdm4 5348 . . . . . . . . 9 dom 𝑥 = ran 𝑥
48 df-rn 5154 . . . . . . . . 9 ran 𝑥 = dom 𝑥
4947, 48uneq12i 3798 . . . . . . . 8 (dom 𝑥 ∪ ran 𝑥) = (ran 𝑥 ∪ dom 𝑥)
5049equncomi 3792 . . . . . . 7 (dom 𝑥 ∪ ran 𝑥) = (dom 𝑥 ∪ ran 𝑥)
5150reseq2i 5425 . . . . . 6 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑥 ∪ ran 𝑥))
5246, 51eqtr2i 2674 . . . . 5 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑥 ∪ ran 𝑥))
53 cnvss 5327 . . . . 5 (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)
5452, 53syl5eqss 3682 . . . 4 (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)
55 dmeq 5356 . . . . . . 7 (𝑦 = 𝑥 → dom 𝑦 = dom 𝑥)
56 rneq 5383 . . . . . . 7 (𝑦 = 𝑥 → ran 𝑦 = ran 𝑥)
5755, 56uneq12d 3801 . . . . . 6 (𝑦 = 𝑥 → (dom 𝑦 ∪ ran 𝑦) = (dom 𝑥 ∪ ran 𝑥))
5857reseq2d 5428 . . . . 5 (𝑦 = 𝑥 → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom 𝑥 ∪ ran 𝑥)))
59 id 22 . . . . 5 (𝑦 = 𝑥𝑦 = 𝑥)
6058, 59sseq12d 3667 . . . 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 481 . 2 ((𝑋𝑉𝑦 = 𝑥) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))
63 dmeq 5356 . . . . 5 (𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) → dom 𝑥 = dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
64 rneq 5383 . . . . 5 (𝑥 = (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) → ran 𝑥 = ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
6563, 64uneq12d 3801 . . . 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 5428 . . 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 3667 . 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 3809 . . 3 𝑋 ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
7069a1i 11 . 2 (𝑋𝑉𝑋 ⊆ (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))))
71 dmexg 7139 . . . . 5 (𝑋𝑉 → dom 𝑋 ∈ V)
72 rnexg 7140 . . . . 5 (𝑋𝑉 → ran 𝑋 ∈ V)
73 unexg 7001 . . . . 5 ((dom 𝑋 ∈ V ∧ ran 𝑋 ∈ V) → (dom 𝑋 ∪ ran 𝑋) ∈ V)
7471, 72, 73syl2anc 694 . . . 4 (𝑋𝑉 → (dom 𝑋 ∪ ran 𝑋) ∈ V)
7574resiexd 6521 . . 3 (𝑋𝑉 → ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ∈ V)
76 unexg 7001 . . 3 ((𝑋𝑉 ∧ ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ∈ V) → (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∈ V)
7775, 76mpdan 703 . 2 (𝑋𝑉 → (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ∈ V)
78 dmun 5363 . . . . . 6 dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) = (dom 𝑋 ∪ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
79 ssun1 3809 . . . . . . 7 dom 𝑋 ⊆ (dom 𝑋 ∪ ran 𝑋)
80 dmresi 5492 . . . . . . . 8 dom ( I ↾ (dom 𝑋 ∪ ran 𝑋)) = (dom 𝑋 ∪ ran 𝑋)
8180eqimssi 3692 . . . . . . 7 dom ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ⊆ (dom 𝑋 ∪ ran 𝑋)
8279, 81unssi 3821 . . . . . 6 (dom 𝑋 ∪ dom ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
8378, 82eqsstri 3668 . . . . 5 dom (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
84 rnun 5576 . . . . . 6 ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) = (ran 𝑋 ∪ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)))
85 ssun2 3810 . . . . . . 7 ran 𝑋 ⊆ (dom 𝑋 ∪ ran 𝑋)
86 rnresi 5514 . . . . . . . 8 ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)) = (dom 𝑋 ∪ ran 𝑋)
8786eqimssi 3692 . . . . . . 7 ran ( I ↾ (dom 𝑋 ∪ ran 𝑋)) ⊆ (dom 𝑋 ∪ ran 𝑋)
8885, 87unssi 3821 . . . . . 6 (ran 𝑋 ∪ ran ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
8984, 88eqsstri 3668 . . . . 5 ran (𝑋 ∪ ( I ↾ (dom 𝑋 ∪ ran 𝑋))) ⊆ (dom 𝑋 ∪ ran 𝑋)
9083, 89pm3.2i 470 . . . 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 3820 . . . . 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 5460 . . . . 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 3812 . . . 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 38247 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 383   = wceq 1523  wcel 2030  {cab 2637  Vcvv 3231  cdif 3604  cun 3605  wss 3607  c0 3948   cint 4507   I cid 5052  ccnv 5142  dom cdm 5143  ran crn 5144  cres 5145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-1st 7210  df-2nd 7211
This theorem is referenced by: (None)
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