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Theorem trclrelexplem 38610
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 6850 . . . 4 (𝑥 = 1 → ((𝐷𝑟𝑘)↑𝑟𝑥) = ((𝐷𝑟𝑘)↑𝑟1))
21iuneq2d 4703 . . 3 (𝑥 = 1 → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) = 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟1))
3 oveq2 6850 . . 3 (𝑥 = 1 → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟1))
42, 3sseq12d 3794 . 2 (𝑥 = 1 → ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) ↔ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟1) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟1)))
5 oveq2 6850 . . . 4 (𝑥 = 𝑦 → ((𝐷𝑟𝑘)↑𝑟𝑥) = ((𝐷𝑟𝑘)↑𝑟𝑦))
65iuneq2d 4703 . . 3 (𝑥 = 𝑦 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) = 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦))
7 oveq2 6850 . . 3 (𝑥 = 𝑦 → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦))
86, 7sseq12d 3794 . 2 (𝑥 = 𝑦 → ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) ↔ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦)))
9 oveq2 6850 . . . 4 (𝑥 = (𝑦 + 1) → ((𝐷𝑟𝑘)↑𝑟𝑥) = ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)))
109iuneq2d 4703 . . 3 (𝑥 = (𝑦 + 1) → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) = 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)))
11 oveq2 6850 . . 3 (𝑥 = (𝑦 + 1) → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟(𝑦 + 1)))
1210, 11sseq12d 3794 . 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 6850 . . . 4 (𝑥 = 𝑁 → ((𝐷𝑟𝑘)↑𝑟𝑥) = ((𝐷𝑟𝑘)↑𝑟𝑁))
1413iuneq2d 4703 . . 3 (𝑥 = 𝑁 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) = 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑁))
15 oveq2 6850 . . 3 (𝑥 = 𝑁 → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑁))
1614, 15sseq12d 3794 . 2 (𝑥 = 𝑁 → ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) ↔ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑁) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑁)))
17 oveq2 6850 . . . . . 6 (𝑘 = 𝑙 → (𝐷𝑟𝑘) = (𝐷𝑟𝑙))
1817cbviunv 4715 . . . . 5 𝑘 ∈ ℕ (𝐷𝑟𝑘) = 𝑙 ∈ ℕ (𝐷𝑟𝑙)
19 oveq2 6850 . . . . . 6 (𝑙 = 𝑗 → (𝐷𝑟𝑙) = (𝐷𝑟𝑗))
2019cbviunv 4715 . . . . 5 𝑙 ∈ ℕ (𝐷𝑟𝑙) = 𝑗 ∈ ℕ (𝐷𝑟𝑗)
2118, 20eqtri 2787 . . . 4 𝑘 ∈ ℕ (𝐷𝑟𝑘) = 𝑗 ∈ ℕ (𝐷𝑟𝑗)
22 ovex 6874 . . . . . 6 (𝐷𝑟𝑘) ∈ V
23 relexp1g 14053 . . . . . 6 ((𝐷𝑟𝑘) ∈ V → ((𝐷𝑟𝑘)↑𝑟1) = (𝐷𝑟𝑘))
2422, 23mp1i 13 . . . . 5 (𝑘 ∈ ℕ → ((𝐷𝑟𝑘)↑𝑟1) = (𝐷𝑟𝑘))
2524iuneq2i 4695 . . . 4 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟1) = 𝑘 ∈ ℕ (𝐷𝑟𝑘)
26 nnex 11281 . . . . . 6 ℕ ∈ V
27 ovex 6874 . . . . . 6 (𝐷𝑟𝑗) ∈ V
2826, 27iunex 7345 . . . . 5 𝑗 ∈ ℕ (𝐷𝑟𝑗) ∈ V
29 relexp1g 14053 . . . . 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 2797 . . 3 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟1) = ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟1)
3231eqimssi 3819 . 2 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟1) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟1)
33 oveq2 6850 . . . . . . . . . 10 (𝑘 = 𝑚 → (𝐷𝑟𝑘) = (𝐷𝑟𝑚))
3433oveq1d 6857 . . . . . . . . 9 (𝑘 = 𝑚 → ((𝐷𝑟𝑘)↑𝑟𝑦) = ((𝐷𝑟𝑚)↑𝑟𝑦))
3534, 33coeq12d 5455 . . . . . . . 8 (𝑘 = 𝑚 → (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)) = (((𝐷𝑟𝑚)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
3635cbviunv 4715 . . . . . . 7 𝑘 ∈ ℕ (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)) = 𝑚 ∈ ℕ (((𝐷𝑟𝑚)↑𝑟𝑦) ∘ (𝐷𝑟𝑚))
37 ss2iun 4692 . . . . . . . 8 (∀𝑚 ∈ ℕ (((𝐷𝑟𝑚)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) ⊆ ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) → 𝑚 ∈ ℕ (((𝐷𝑟𝑚)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) ⊆ 𝑚 ∈ ℕ ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
3834ssiun2s 4720 . . . . . . . . 9 (𝑚 ∈ ℕ → ((𝐷𝑟𝑚)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦))
39 coss1 5446 . . . . . . . . 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 3073 . . . . . . 7 𝑚 ∈ ℕ (((𝐷𝑟𝑚)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) ⊆ 𝑚 ∈ ℕ ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚))
4236, 41eqsstri 3795 . . . . . 6 𝑘 ∈ ℕ (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)) ⊆ 𝑚 ∈ ℕ ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚))
43 coss1 5446 . . . . . . . 8 ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) → ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) ⊆ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
4443ralrimivw 3114 . . . . . . 7 ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) → ∀𝑚 ∈ ℕ ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) ⊆ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
45 ss2iun 4692 . . . . . . 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, 46syl5ss 3772 . . . . 5 ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) → 𝑘 ∈ ℕ (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)) ⊆ 𝑚 ∈ ℕ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
4847adantl 473 . . . 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 14052 . . . . . . 7 (((𝐷𝑟𝑘) ∈ V ∧ 𝑦 ∈ ℕ) → ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)) = (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)))
5022, 49mpan 681 . . . . . 6 (𝑦 ∈ ℕ → ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)) = (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)))
5150iuneq2d 4703 . . . . 5 (𝑦 ∈ ℕ → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)) = 𝑘 ∈ ℕ (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)))
5251adantr 472 . . . 4 ((𝑦 ∈ ℕ ∧ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦)) → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)) = 𝑘 ∈ ℕ (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)))
53 relexpsucnnr 14052 . . . . . . 7 (( 𝑗 ∈ ℕ (𝐷𝑟𝑗) ∈ V ∧ 𝑦 ∈ ℕ) → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟(𝑦 + 1)) = (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝐷𝑟𝑗)))
5428, 53mpan 681 . . . . . 6 (𝑦 ∈ ℕ → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟(𝑦 + 1)) = (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝐷𝑟𝑗)))
55 oveq2 6850 . . . . . . . . 9 (𝑗 = 𝑚 → (𝐷𝑟𝑗) = (𝐷𝑟𝑚))
5655cbviunv 4715 . . . . . . . 8 𝑗 ∈ ℕ (𝐷𝑟𝑗) = 𝑚 ∈ ℕ (𝐷𝑟𝑚)
5756coeq2i 5451 . . . . . . 7 (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝐷𝑟𝑗)) = (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑚 ∈ ℕ (𝐷𝑟𝑚))
58 coiun 5831 . . . . . . 7 (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑚 ∈ ℕ (𝐷𝑟𝑚)) = 𝑚 ∈ ℕ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚))
5957, 58eqtri 2787 . . . . . 6 (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝐷𝑟𝑗)) = 𝑚 ∈ ℕ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚))
6054, 59syl6eq 2815 . . . . 5 (𝑦 ∈ ℕ → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟(𝑦 + 1)) = 𝑚 ∈ ℕ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
6160adantr 472 . . . 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 3808 . . 3 ((𝑦 ∈ ℕ ∧ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦)) → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟(𝑦 + 1)))
6362ex 401 . 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 11294 1 (𝑁 ∈ ℕ → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑁) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  wss 3732   ciun 4676  ccom 5281  (class class class)co 6842  1c1 10190   + caddc 10192  cn 11274  𝑟crelexp 14047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-n0 11539  df-z 11625  df-uz 11887  df-seq 13009  df-relexp 14048
This theorem is referenced by: (None)
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