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Theorem trclrelexplem 43701
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 7439 . . . 4 (𝑥 = 1 → ((𝐷𝑟𝑘)↑𝑟𝑥) = ((𝐷𝑟𝑘)↑𝑟1))
21iuneq2d 5027 . . 3 (𝑥 = 1 → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) = 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟1))
3 oveq2 7439 . . 3 (𝑥 = 1 → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟1))
42, 3sseq12d 4029 . 2 (𝑥 = 1 → ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) ↔ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟1) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟1)))
5 oveq2 7439 . . . 4 (𝑥 = 𝑦 → ((𝐷𝑟𝑘)↑𝑟𝑥) = ((𝐷𝑟𝑘)↑𝑟𝑦))
65iuneq2d 5027 . . 3 (𝑥 = 𝑦 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) = 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦))
7 oveq2 7439 . . 3 (𝑥 = 𝑦 → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦))
86, 7sseq12d 4029 . 2 (𝑥 = 𝑦 → ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) ↔ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦)))
9 oveq2 7439 . . . 4 (𝑥 = (𝑦 + 1) → ((𝐷𝑟𝑘)↑𝑟𝑥) = ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)))
109iuneq2d 5027 . . 3 (𝑥 = (𝑦 + 1) → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) = 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)))
11 oveq2 7439 . . 3 (𝑥 = (𝑦 + 1) → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟(𝑦 + 1)))
1210, 11sseq12d 4029 . 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 7439 . . . 4 (𝑥 = 𝑁 → ((𝐷𝑟𝑘)↑𝑟𝑥) = ((𝐷𝑟𝑘)↑𝑟𝑁))
1413iuneq2d 5027 . . 3 (𝑥 = 𝑁 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) = 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑁))
15 oveq2 7439 . . 3 (𝑥 = 𝑁 → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑁))
1614, 15sseq12d 4029 . 2 (𝑥 = 𝑁 → ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑥) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑥) ↔ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑁) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑁)))
17 oveq2 7439 . . . . . 6 (𝑘 = 𝑙 → (𝐷𝑟𝑘) = (𝐷𝑟𝑙))
1817cbviunv 5045 . . . . 5 𝑘 ∈ ℕ (𝐷𝑟𝑘) = 𝑙 ∈ ℕ (𝐷𝑟𝑙)
19 oveq2 7439 . . . . . 6 (𝑙 = 𝑗 → (𝐷𝑟𝑙) = (𝐷𝑟𝑗))
2019cbviunv 5045 . . . . 5 𝑙 ∈ ℕ (𝐷𝑟𝑙) = 𝑗 ∈ ℕ (𝐷𝑟𝑗)
2118, 20eqtri 2763 . . . 4 𝑘 ∈ ℕ (𝐷𝑟𝑘) = 𝑗 ∈ ℕ (𝐷𝑟𝑗)
22 ovex 7464 . . . . . 6 (𝐷𝑟𝑘) ∈ V
23 relexp1g 15062 . . . . . 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 12270 . . . . . 6 ℕ ∈ V
27 ovex 7464 . . . . . 6 (𝐷𝑟𝑗) ∈ V
2826, 27iunex 7992 . . . . 5 𝑗 ∈ ℕ (𝐷𝑟𝑗) ∈ V
29 relexp1g 15062 . . . . 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 2773 . . 3 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟1) = ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟1)
3231eqimssi 4056 . 2 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟1) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟1)
33 oveq2 7439 . . . . . . . . . 10 (𝑘 = 𝑚 → (𝐷𝑟𝑘) = (𝐷𝑟𝑚))
3433oveq1d 7446 . . . . . . . . 9 (𝑘 = 𝑚 → ((𝐷𝑟𝑘)↑𝑟𝑦) = ((𝐷𝑟𝑚)↑𝑟𝑦))
3534, 33coeq12d 5878 . . . . . . . 8 (𝑘 = 𝑚 → (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)) = (((𝐷𝑟𝑚)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
3635cbviunv 5045 . . . . . . 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 5053 . . . . . . . . 9 (𝑚 ∈ ℕ → ((𝐷𝑟𝑚)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦))
39 coss1 5869 . . . . . . . . 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 4030 . . . . . 6 𝑘 ∈ ℕ (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)) ⊆ 𝑚 ∈ ℕ ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚))
43 coss1 5869 . . . . . . . 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 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 4007 . . . . 5 ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) → 𝑘 ∈ ℕ (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)) ⊆ 𝑚 ∈ ℕ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
4847adantl 481 . . . 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 15061 . . . . . . 7 (((𝐷𝑟𝑘) ∈ V ∧ 𝑦 ∈ ℕ) → ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)) = (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)))
5022, 49mpan 690 . . . . . 6 (𝑦 ∈ ℕ → ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)) = (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)))
5150iuneq2d 5027 . . . . 5 (𝑦 ∈ ℕ → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)) = 𝑘 ∈ ℕ (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)))
5251adantr 480 . . . 4 ((𝑦 ∈ ℕ ∧ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦)) → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)) = 𝑘 ∈ ℕ (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)))
53 relexpsucnnr 15061 . . . . . . 7 (( 𝑗 ∈ ℕ (𝐷𝑟𝑗) ∈ V ∧ 𝑦 ∈ ℕ) → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟(𝑦 + 1)) = (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝐷𝑟𝑗)))
5428, 53mpan 690 . . . . . 6 (𝑦 ∈ ℕ → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟(𝑦 + 1)) = (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝐷𝑟𝑗)))
55 oveq2 7439 . . . . . . . . 9 (𝑗 = 𝑚 → (𝐷𝑟𝑗) = (𝐷𝑟𝑚))
5655cbviunv 5045 . . . . . . . 8 𝑗 ∈ ℕ (𝐷𝑟𝑗) = 𝑚 ∈ ℕ (𝐷𝑟𝑚)
5756coeq2i 5874 . . . . . . 7 (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝐷𝑟𝑗)) = (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑚 ∈ ℕ (𝐷𝑟𝑚))
58 coiun 6278 . . . . . . 7 (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑚 ∈ ℕ (𝐷𝑟𝑚)) = 𝑚 ∈ ℕ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚))
5957, 58eqtri 2763 . . . . . 6 (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝐷𝑟𝑗)) = 𝑚 ∈ ℕ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚))
6054, 59eqtrdi 2791 . . . . 5 (𝑦 ∈ ℕ → ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟(𝑦 + 1)) = 𝑚 ∈ ℕ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
6160adantr 480 . . . 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 4043 . . 3 ((𝑦 ∈ ℕ ∧ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦)) → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟(𝑦 + 1)) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟(𝑦 + 1)))
6362ex 412 . 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 12282 1 (𝑁 ∈ ℕ → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑁) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  wss 3963   ciun 4996  ccom 5693  (class class class)co 7431  1c1 11154   + caddc 11156  cn 12264  𝑟crelexp 15055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-seq 14040  df-relexp 15056
This theorem is referenced by: (None)
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