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Theorem trclrelexplem 43280
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 7427 . . . 4 (𝑥 = 1 → ((𝐷𝑟𝑘)↑𝑟𝑥) = ((𝐷𝑟𝑘)↑𝑟1))
21iuneq2d 5026 . . 3 (𝑥 = 1 → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) = 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟1))
3 oveq2 7427 . . 3 (𝑥 = 1 → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟1))
42, 3sseq12d 4010 . 2 (𝑥 = 1 → ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) ↔ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟1) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟1)))
5 oveq2 7427 . . . 4 (𝑥 = 𝑦 → ((𝐷𝑟𝑘)↑𝑟𝑥) = ((𝐷𝑟𝑘)↑𝑟𝑦))
65iuneq2d 5026 . . 3 (𝑥 = 𝑦 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) = 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦))
7 oveq2 7427 . . 3 (𝑥 = 𝑦 → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦))
86, 7sseq12d 4010 . 2 (𝑥 = 𝑦 → ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) ↔ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦)))
9 oveq2 7427 . . . 4 (𝑥 = (𝑦 + 1) → ((𝐷𝑟𝑘)↑𝑟𝑥) = ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)))
109iuneq2d 5026 . . 3 (𝑥 = (𝑦 + 1) → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) = 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)))
11 oveq2 7427 . . 3 (𝑥 = (𝑦 + 1) → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟(𝑦 + 1)))
1210, 11sseq12d 4010 . 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 7427 . . . 4 (𝑥 = 𝑁 → ((𝐷𝑟𝑘)↑𝑟𝑥) = ((𝐷𝑟𝑘)↑𝑟𝑁))
1413iuneq2d 5026 . . 3 (𝑥 = 𝑁 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) = 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑁))
15 oveq2 7427 . . 3 (𝑥 = 𝑁 → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑁))
1614, 15sseq12d 4010 . 2 (𝑥 = 𝑁 → ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) ↔ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑁) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑁)))
17 oveq2 7427 . . . . . 6 (𝑘 = 𝑙 → (𝐷𝑟𝑘) = (𝐷𝑟𝑙))
1817cbviunv 5044 . . . . 5 𝑘 ∈ ℕ (𝐷𝑟𝑘) = 𝑙 ∈ ℕ (𝐷𝑟𝑙)
19 oveq2 7427 . . . . . 6 (𝑙 = 𝑗 → (𝐷𝑟𝑙) = (𝐷𝑟𝑗))
2019cbviunv 5044 . . . . 5 𝑙 ∈ ℕ (𝐷𝑟𝑙) = 𝑗 ∈ ℕ (𝐷𝑟𝑗)
2118, 20eqtri 2753 . . . 4 𝑘 ∈ ℕ (𝐷𝑟𝑘) = 𝑗 ∈ ℕ (𝐷𝑟𝑗)
22 ovex 7452 . . . . . 6 (𝐷𝑟𝑘) ∈ V
23 relexp1g 15009 . . . . . 6 ((𝐷𝑟𝑘) ∈ V → ((𝐷𝑟𝑘)↑𝑟1) = (𝐷𝑟𝑘))
2422, 23mp1i 13 . . . . 5 (𝑘 ∈ ℕ → ((𝐷𝑟𝑘)↑𝑟1) = (𝐷𝑟𝑘))
2524iuneq2i 5018 . . . 4 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟1) = 𝑘 ∈ ℕ (𝐷𝑟𝑘)
26 nnex 12251 . . . . . 6 ℕ ∈ V
27 ovex 7452 . . . . . 6 (𝐷𝑟𝑗) ∈ V
2826, 27iunex 7973 . . . . 5 𝑗 ∈ ℕ (𝐷𝑟𝑗) ∈ V
29 relexp1g 15009 . . . . 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 2763 . . 3 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟1) = ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟1)
3231eqimssi 4037 . 2 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟1) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟1)
33 oveq2 7427 . . . . . . . . . 10 (𝑘 = 𝑚 → (𝐷𝑟𝑘) = (𝐷𝑟𝑚))
3433oveq1d 7434 . . . . . . . . 9 (𝑘 = 𝑚 → ((𝐷𝑟𝑘)↑𝑟𝑦) = ((𝐷𝑟𝑚)↑𝑟𝑦))
3534, 33coeq12d 5867 . . . . . . . 8 (𝑘 = 𝑚 → (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)) = (((𝐷𝑟𝑚)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
3635cbviunv 5044 . . . . . . 7 𝑘 ∈ ℕ (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)) = 𝑚 ∈ ℕ (((𝐷𝑟𝑚)↑𝑟𝑦) ∘ (𝐷𝑟𝑚))
37 ss2iun 5015 . . . . . . . 8 (∀𝑚 ∈ ℕ (((𝐷𝑟𝑚)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) ⊆ ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) → 𝑚 ∈ ℕ (((𝐷𝑟𝑚)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) ⊆ 𝑚 ∈ ℕ ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
3834ssiun2s 5052 . . . . . . . . 9 (𝑚 ∈ ℕ → ((𝐷𝑟𝑚)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦))
39 coss1 5858 . . . . . . . . 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 3056 . . . . . . 7 𝑚 ∈ ℕ (((𝐷𝑟𝑚)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) ⊆ 𝑚 ∈ ℕ ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚))
4236, 41eqsstri 4011 . . . . . 6 𝑘 ∈ ℕ (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)) ⊆ 𝑚 ∈ ℕ ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚))
43 coss1 5858 . . . . . . . 8 ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) → ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) ⊆ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
4443ralrimivw 3139 . . . . . . 7 ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) → ∀𝑚 ∈ ℕ ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) ⊆ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
45 ss2iun 5015 . . . . . . 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 3988 . . . . 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 15008 . . . . . . 7 (((𝐷𝑟𝑘) ∈ V ∧ 𝑦 ∈ ℕ) → ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)) = (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)))
5022, 49mpan 688 . . . . . 6 (𝑦 ∈ ℕ → ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)) = (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)))
5150iuneq2d 5026 . . . . 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 15008 . . . . . . 7 (( 𝑗 ∈ ℕ (𝐷𝑟𝑗) ∈ V ∧ 𝑦 ∈ ℕ) → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟(𝑦 + 1)) = (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝐷𝑟𝑗)))
5428, 53mpan 688 . . . . . 6 (𝑦 ∈ ℕ → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟(𝑦 + 1)) = (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝐷𝑟𝑗)))
55 oveq2 7427 . . . . . . . . 9 (𝑗 = 𝑚 → (𝐷𝑟𝑗) = (𝐷𝑟𝑚))
5655cbviunv 5044 . . . . . . . 8 𝑗 ∈ ℕ (𝐷𝑟𝑗) = 𝑚 ∈ ℕ (𝐷𝑟𝑚)
5756coeq2i 5863 . . . . . . 7 (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝐷𝑟𝑗)) = (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑚 ∈ ℕ (𝐷𝑟𝑚))
58 coiun 6262 . . . . . . 7 (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑚 ∈ ℕ (𝐷𝑟𝑚)) = 𝑚 ∈ ℕ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚))
5957, 58eqtri 2753 . . . . . 6 (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝐷𝑟𝑗)) = 𝑚 ∈ ℕ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚))
6054, 59eqtrdi 2781 . . . . 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 4024 . . 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 12263 1 (𝑁 ∈ ℕ → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑁) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3050  Vcvv 3461  wss 3944   ciun 4997  ccom 5682  (class class class)co 7419  1c1 11141   + caddc 11143  cn 12245  𝑟crelexp 15002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-n0 12506  df-z 12592  df-uz 12856  df-seq 14003  df-relexp 15003
This theorem is referenced by: (None)
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