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Theorem trclrelexplem 43724
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 5022 . . 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 4017 . 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 5022 . . 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 4017 . 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 5022 . . 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 4017 . 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 5022 . . 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 4017 . 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 5040 . . . . 5 𝑘 ∈ ℕ (𝐷𝑟𝑘) = 𝑙 ∈ ℕ (𝐷𝑟𝑙)
19 oveq2 7439 . . . . . 6 (𝑙 = 𝑗 → (𝐷𝑟𝑙) = (𝐷𝑟𝑗))
2019cbviunv 5040 . . . . 5 𝑙 ∈ ℕ (𝐷𝑟𝑙) = 𝑗 ∈ ℕ (𝐷𝑟𝑗)
2118, 20eqtri 2765 . . . 4 𝑘 ∈ ℕ (𝐷𝑟𝑘) = 𝑗 ∈ ℕ (𝐷𝑟𝑗)
22 ovex 7464 . . . . . 6 (𝐷𝑟𝑘) ∈ V
23 relexp1g 15065 . . . . . 6 ((𝐷𝑟𝑘) ∈ V → ((𝐷𝑟𝑘)↑𝑟1) = (𝐷𝑟𝑘))
2422, 23mp1i 13 . . . . 5 (𝑘 ∈ ℕ → ((𝐷𝑟𝑘)↑𝑟1) = (𝐷𝑟𝑘))
2524iuneq2i 5013 . . . 4 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟1) = 𝑘 ∈ ℕ (𝐷𝑟𝑘)
26 nnex 12272 . . . . . 6 ℕ ∈ V
27 ovex 7464 . . . . . 6 (𝐷𝑟𝑗) ∈ V
2826, 27iunex 7993 . . . . 5 𝑗 ∈ ℕ (𝐷𝑟𝑗) ∈ V
29 relexp1g 15065 . . . . 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 2775 . . 3 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟1) = ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟1)
3231eqimssi 4044 . 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 5875 . . . . . . . 8 (𝑘 = 𝑚 → (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)) = (((𝐷𝑟𝑚)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
3635cbviunv 5040 . . . . . . 7 𝑘 ∈ ℕ (((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑘)) = 𝑚 ∈ ℕ (((𝐷𝑟𝑚)↑𝑟𝑦) ∘ (𝐷𝑟𝑚))
37 ss2iun 5010 . . . . . . . 8 (∀𝑚 ∈ ℕ (((𝐷𝑟𝑚)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) ⊆ ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) → 𝑚 ∈ ℕ (((𝐷𝑟𝑚)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) ⊆ 𝑚 ∈ ℕ ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
3834ssiun2s 5048 . . . . . . . . 9 (𝑚 ∈ ℕ → ((𝐷𝑟𝑚)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦))
39 coss1 5866 . . . . . . . . 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 3067 . . . . . . 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 5866 . . . . . . . 8 ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) → ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) ⊆ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
4443ralrimivw 3150 . . . . . . 7 ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) → ∀𝑚 ∈ ℕ ( 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)) ⊆ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚)))
45 ss2iun 5010 . . . . . . 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 3995 . . . . 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 15064 . . . . . . 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 5022 . . . . 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 15064 . . . . . . 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 5040 . . . . . . . 8 𝑗 ∈ ℕ (𝐷𝑟𝑗) = 𝑚 ∈ ℕ (𝐷𝑟𝑚)
5756coeq2i 5871 . . . . . . 7 (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝐷𝑟𝑗)) = (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑚 ∈ ℕ (𝐷𝑟𝑚))
58 coiun 6276 . . . . . . 7 (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑚 ∈ ℕ (𝐷𝑟𝑚)) = 𝑚 ∈ ℕ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚))
5957, 58eqtri 2765 . . . . . 6 (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝐷𝑟𝑗)) = 𝑚 ∈ ℕ (( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑦) ∘ (𝐷𝑟𝑚))
6054, 59eqtrdi 2793 . . . . 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 4039 . . 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 12284 1 (𝑁 ∈ ℕ → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑁) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  wss 3951   ciun 4991  ccom 5689  (class class class)co 7431  1c1 11156   + caddc 11158  cn 12266  𝑟crelexp 15058
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-seq 14043  df-relexp 15059
This theorem is referenced by: (None)
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