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Theorem trclrelexplem 42764
Description: The union of relational powers to positive multiples of 𝑁 is a subset to the transitive closure raised to the power of 𝑁. (Contributed by RP, 15-Jun-2020.)
Assertion
Ref Expression
trclrelexplem (𝑁 ∈ ℕ → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑁) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑁))
Distinct variable groups:   𝐷,𝑗   𝐷,𝑘   𝑘,𝑁
Allowed substitution hint:   𝑁(𝑗)

Proof of Theorem trclrelexplem
Dummy variables 𝑥 𝑦 𝑙 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7419 . . . 4 (𝑥 = 1 → ((𝐷𝑟𝑘)↑𝑟𝑥) = ((𝐷𝑟𝑘)↑𝑟1))
21iuneq2d 5025 . . 3 (𝑥 = 1 → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) = 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟1))
3 oveq2 7419 . . 3 (𝑥 = 1 → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟1))
42, 3sseq12d 4014 . 2 (𝑥 = 1 → ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) ↔ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟1) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟1)))
5 oveq2 7419 . . . 4 (𝑥 = 𝑦 → ((𝐷𝑟𝑘)↑𝑟𝑥) = ((𝐷𝑟𝑘)↑𝑟𝑦))
65iuneq2d 5025 . . 3 (𝑥 = 𝑦 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) = 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦))
7 oveq2 7419 . . 3 (𝑥 = 𝑦 → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦))
86, 7sseq12d 4014 . 2 (𝑥 = 𝑦 → ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) ↔ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦)))
9 oveq2 7419 . . . 4 (𝑥 = (𝑦 + 1) → ((𝐷𝑟𝑘)↑𝑟𝑥) = ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)))
109iuneq2d 5025 . . 3 (𝑥 = (𝑦 + 1) → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) = 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)))
11 oveq2 7419 . . 3 (𝑥 = (𝑦 + 1) → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟(𝑦 + 1)))
1210, 11sseq12d 4014 . 2 (𝑥 = (𝑦 + 1) → ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) ↔ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟(𝑦 + 1))))
13 oveq2 7419 . . . 4 (𝑥 = 𝑁 → ((𝐷𝑟𝑘)↑𝑟𝑥) = ((𝐷𝑟𝑘)↑𝑟𝑁))
1413iuneq2d 5025 . . 3 (𝑥 = 𝑁 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) = 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑁))
15 oveq2 7419 . . 3 (𝑥 = 𝑁 → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑁))
1614, 15sseq12d 4014 . 2 (𝑥 = 𝑁 → ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) ↔ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑁) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑁)))
17 oveq2 7419 . . . . . 6 (𝑘 = 𝑙 → (𝐷𝑟𝑘) = (𝐷𝑟𝑙))
1817cbviunv 5042 . . . . 5 𝑘 ∈ ℕ (𝐷𝑟𝑘) = 𝑙 ∈ ℕ (𝐷𝑟𝑙)
19 oveq2 7419 . . . . . 6 (𝑙 = 𝑗 → (𝐷𝑟𝑙) = (𝐷𝑟𝑗))
2019cbviunv 5042 . . . . 5 𝑙 ∈ ℕ (𝐷𝑟𝑙) = 𝑗 ∈ ℕ (𝐷𝑟𝑗)
2118, 20eqtri 2758 . . . 4 𝑘 ∈ ℕ (𝐷𝑟𝑘) = 𝑗 ∈ ℕ (𝐷𝑟𝑗)
22 ovex 7444 . . . . . 6 (𝐷𝑟𝑘) ∈ V
23 relexp1g 14977 . . . . . 6 ((𝐷𝑟𝑘) ∈ V → ((𝐷𝑟𝑘)↑𝑟1) = (𝐷𝑟𝑘))
2422, 23mp1i 13 . . . . 5 (𝑘 ∈ ℕ → ((𝐷𝑟𝑘)↑𝑟1) = (𝐷𝑟𝑘))
2524iuneq2i 5017 . . . 4 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟1) = 𝑘 ∈ ℕ (𝐷𝑟𝑘)
26 nnex 12222 . . . . . 6 ℕ ∈ V
27 ovex 7444 . . . . . 6 (𝐷𝑟𝑗) ∈ V
2826, 27iunex 7957 . . . . 5 𝑗 ∈ ℕ (𝐷𝑟𝑗) ∈ V
29 relexp1g 14977 . . . . 5 ( 𝑗 ∈ ℕ (𝐷𝑟𝑗) ∈ V → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟1) = 𝑗 ∈ ℕ (𝐷𝑟𝑗))
3028, 29ax-mp 5 . . . 4 ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟1) = 𝑗 ∈ ℕ (𝐷𝑟𝑗)
3121, 25, 303eqtr4i 2768 . . 3 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟1) = ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟1)
3231eqimssi 4041 . 2 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟1) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟1)
33 oveq2 7419 . . . . . . . . . 10 (𝑘 = 𝑚 → (𝐷𝑟𝑘) = (𝐷𝑟𝑚))
3433oveq1d 7426 . . . . . . . . 9 (𝑘 = 𝑚 → ((𝐷𝑟𝑘)↑𝑟𝑦) = ((𝐷𝑟𝑚)↑𝑟𝑦))
3534, 33coeq12d 5863 . . . . . . . 8 (𝑘 = 𝑚 → (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)) = (((𝐷𝑟𝑚)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
3635cbviunv 5042 . . . . . . 7 𝑘 ∈ ℕ (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)) = 𝑚 ∈ ℕ (((𝐷𝑟𝑚)↑𝑟𝑦) ∘ (𝐷𝑟𝑚))
37 ss2iun 5014 . . . . . . . 8 (∀𝑚 ∈ ℕ (((𝐷𝑟𝑚)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) ⊆ ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) → 𝑚 ∈ ℕ (((𝐷𝑟𝑚)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) ⊆ 𝑚 ∈ ℕ ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
3834ssiun2s 5050 . . . . . . . . 9 (𝑚 ∈ ℕ → ((𝐷𝑟𝑚)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦))
39 coss1 5854 . . . . . . . . 9 (((𝐷𝑟𝑚)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) → (((𝐷𝑟𝑚)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) ⊆ ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
4038, 39syl 17 . . . . . . . 8 (𝑚 ∈ ℕ → (((𝐷𝑟𝑚)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) ⊆ ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
4137, 40mprg 3065 . . . . . . 7 𝑚 ∈ ℕ (((𝐷𝑟𝑚)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) ⊆ 𝑚 ∈ ℕ ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚))
4236, 41eqsstri 4015 . . . . . 6 𝑘 ∈ ℕ (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)) ⊆ 𝑚 ∈ ℕ ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚))
43 coss1 5854 . . . . . . . 8 ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) → ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) ⊆ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
4443ralrimivw 3148 . . . . . . 7 ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) → ∀𝑚 ∈ ℕ ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) ⊆ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
45 ss2iun 5014 . . . . . . 7 (∀𝑚 ∈ ℕ ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) ⊆ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) → 𝑚 ∈ ℕ ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) ⊆ 𝑚 ∈ ℕ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
4644, 45syl 17 . . . . . 6 ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) → 𝑚 ∈ ℕ ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) ⊆ 𝑚 ∈ ℕ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
4742, 46sstrid 3992 . . . . 5 ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) → 𝑘 ∈ ℕ (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)) ⊆ 𝑚 ∈ ℕ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
4847adantl 480 . . . 4 ((𝑦 ∈ ℕ ∧ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦)) → 𝑘 ∈ ℕ (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)) ⊆ 𝑚 ∈ ℕ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
49 relexpsucnnr 14976 . . . . . . 7 (((𝐷𝑟𝑘) ∈ V ∧ 𝑦 ∈ ℕ) → ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)) = (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)))
5022, 49mpan 686 . . . . . 6 (𝑦 ∈ ℕ → ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)) = (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)))
5150iuneq2d 5025 . . . . 5 (𝑦 ∈ ℕ → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)) = 𝑘 ∈ ℕ (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)))
5251adantr 479 . . . 4 ((𝑦 ∈ ℕ ∧ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦)) → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)) = 𝑘 ∈ ℕ (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)))
53 relexpsucnnr 14976 . . . . . . 7 (( 𝑗 ∈ ℕ (𝐷𝑟𝑗) ∈ V ∧ 𝑦 ∈ ℕ) → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟(𝑦 + 1)) = (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝐷𝑟𝑗)))
5428, 53mpan 686 . . . . . 6 (𝑦 ∈ ℕ → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟(𝑦 + 1)) = (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝐷𝑟𝑗)))
55 oveq2 7419 . . . . . . . . 9 (𝑗 = 𝑚 → (𝐷𝑟𝑗) = (𝐷𝑟𝑚))
5655cbviunv 5042 . . . . . . . 8 𝑗 ∈ ℕ (𝐷𝑟𝑗) = 𝑚 ∈ ℕ (𝐷𝑟𝑚)
5756coeq2i 5859 . . . . . . 7 (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝐷𝑟𝑗)) = (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑚 ∈ ℕ (𝐷𝑟𝑚))
58 coiun 6254 . . . . . . 7 (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑚 ∈ ℕ (𝐷𝑟𝑚)) = 𝑚 ∈ ℕ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚))
5957, 58eqtri 2758 . . . . . 6 (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝐷𝑟𝑗)) = 𝑚 ∈ ℕ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚))
6054, 59eqtrdi 2786 . . . . 5 (𝑦 ∈ ℕ → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟(𝑦 + 1)) = 𝑚 ∈ ℕ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
6160adantr 479 . . . 4 ((𝑦 ∈ ℕ ∧ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦)) → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟(𝑦 + 1)) = 𝑚 ∈ ℕ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
6248, 52, 613sstr4d 4028 . . 3 ((𝑦 ∈ ℕ ∧ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦)) → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟(𝑦 + 1)))
6362ex 411 . 2 (𝑦 ∈ ℕ → ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟(𝑦 + 1))))
644, 8, 12, 16, 32, 63nnind 12234 1 (𝑁 ∈ ℕ → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑁) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  wral 3059  Vcvv 3472  wss 3947   ciun 4996  ccom 5679  (class class class)co 7411  1c1 11113   + caddc 11115  cn 12216  𝑟crelexp 14970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-seq 13971  df-relexp 14971
This theorem is referenced by: (None)
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