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Theorem fsuppmapnn0fiubex 12775
Description: If all functions of a finite set of functions over the nonnegative integers are finitely supported, then the support of all these functions is contained in a finite set of sequential integers starting at 0. (Contributed by AV, 2-Oct-2019.)
Assertion
Ref Expression
fsuppmapnn0fiubex ((𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉) → (∀𝑓𝑀 𝑓 finSupp 𝑍 → ∃𝑚 ∈ ℕ0𝑓𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑚)))
Distinct variable groups:   𝑓,𝑀,𝑚   𝑅,𝑓,𝑚   𝑓,𝑉,𝑚   𝑓,𝑍,𝑚

Proof of Theorem fsuppmapnn0fiubex
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 0nn0 11292 . . . . 5 0 ∈ ℕ0
21a1i 11 . . . 4 ((∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) → 0 ∈ ℕ0)
3 oveq2 6643 . . . . . . 7 (𝑚 = 0 → (0...𝑚) = (0...0))
43sseq2d 3625 . . . . . 6 (𝑚 = 0 → ((𝑓 supp 𝑍) ⊆ (0...𝑚) ↔ (𝑓 supp 𝑍) ⊆ (0...0)))
54ralbidv 2983 . . . . 5 (𝑚 = 0 → (∀𝑓𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑚) ↔ ∀𝑓𝑀 (𝑓 supp 𝑍) ⊆ (0...0)))
65adantl 482 . . . 4 (((∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) ∧ 𝑚 = 0) → (∀𝑓𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑚) ↔ ∀𝑓𝑀 (𝑓 supp 𝑍) ⊆ (0...0)))
7 ral0 4067 . . . . . 6 𝑓 ∈ ∅ (𝑓 supp 𝑍) ⊆ (0...0)
8 raleq 3133 . . . . . 6 (∅ = 𝑀 → (∀𝑓 ∈ ∅ (𝑓 supp 𝑍) ⊆ (0...0) ↔ ∀𝑓𝑀 (𝑓 supp 𝑍) ⊆ (0...0)))
97, 8mpbii 223 . . . . 5 (∅ = 𝑀 → ∀𝑓𝑀 (𝑓 supp 𝑍) ⊆ (0...0))
10 0ss 3963 . . . . . . 7 ∅ ⊆ (0...0)
11 sseq1 3618 . . . . . . 7 ((𝑓 supp 𝑍) = ∅ → ((𝑓 supp 𝑍) ⊆ (0...0) ↔ ∅ ⊆ (0...0)))
1210, 11mpbiri 248 . . . . . 6 ((𝑓 supp 𝑍) = ∅ → (𝑓 supp 𝑍) ⊆ (0...0))
1312ralimi 2949 . . . . 5 (∀𝑓𝑀 (𝑓 supp 𝑍) = ∅ → ∀𝑓𝑀 (𝑓 supp 𝑍) ⊆ (0...0))
149, 13jaoi 394 . . . 4 ((∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) → ∀𝑓𝑀 (𝑓 supp 𝑍) ⊆ (0...0))
152, 6, 14rspcedvd 3312 . . 3 ((∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) → ∃𝑚 ∈ ℕ0𝑓𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑚))
16152a1d 26 . 2 ((∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) → ((𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉) → (∀𝑓𝑀 𝑓 finSupp 𝑍 → ∃𝑚 ∈ ℕ0𝑓𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑚))))
17 simplr 791 . . . . 5 (((¬ (∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉)) ∧ ∀𝑓𝑀 𝑓 finSupp 𝑍) → (𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉))
18 simpr 477 . . . . . 6 (((¬ (∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉)) ∧ ∀𝑓𝑀 𝑓 finSupp 𝑍) → ∀𝑓𝑀 𝑓 finSupp 𝑍)
19 ioran 511 . . . . . . . . . 10 (¬ (∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) ↔ (¬ ∅ = 𝑀 ∧ ¬ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅))
20 oveq1 6642 . . . . . . . . . . . . . 14 (𝑓 = 𝑔 → (𝑓 supp 𝑍) = (𝑔 supp 𝑍))
2120eqeq1d 2622 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → ((𝑓 supp 𝑍) = ∅ ↔ (𝑔 supp 𝑍) = ∅))
2221cbvralv 3166 . . . . . . . . . . . 12 (∀𝑓𝑀 (𝑓 supp 𝑍) = ∅ ↔ ∀𝑔𝑀 (𝑔 supp 𝑍) = ∅)
2322notbii 310 . . . . . . . . . . 11 (¬ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅ ↔ ¬ ∀𝑔𝑀 (𝑔 supp 𝑍) = ∅)
2423anbi2i 729 . . . . . . . . . 10 ((¬ ∅ = 𝑀 ∧ ¬ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) ↔ (¬ ∅ = 𝑀 ∧ ¬ ∀𝑔𝑀 (𝑔 supp 𝑍) = ∅))
2519, 24bitri 264 . . . . . . . . 9 (¬ (∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) ↔ (¬ ∅ = 𝑀 ∧ ¬ ∀𝑔𝑀 (𝑔 supp 𝑍) = ∅))
26 rexnal 2992 . . . . . . . . . . 11 (∃𝑔𝑀 ¬ (𝑔 supp 𝑍) = ∅ ↔ ¬ ∀𝑔𝑀 (𝑔 supp 𝑍) = ∅)
27 df-ne 2792 . . . . . . . . . . . . 13 ((𝑔 supp 𝑍) ≠ ∅ ↔ ¬ (𝑔 supp 𝑍) = ∅)
2827bicomi 214 . . . . . . . . . . . 12 (¬ (𝑔 supp 𝑍) = ∅ ↔ (𝑔 supp 𝑍) ≠ ∅)
2928rexbii 3037 . . . . . . . . . . 11 (∃𝑔𝑀 ¬ (𝑔 supp 𝑍) = ∅ ↔ ∃𝑔𝑀 (𝑔 supp 𝑍) ≠ ∅)
3026, 29sylbb1 227 . . . . . . . . . 10 (¬ ∀𝑔𝑀 (𝑔 supp 𝑍) = ∅ → ∃𝑔𝑀 (𝑔 supp 𝑍) ≠ ∅)
3130adantl 482 . . . . . . . . 9 ((¬ ∅ = 𝑀 ∧ ¬ ∀𝑔𝑀 (𝑔 supp 𝑍) = ∅) → ∃𝑔𝑀 (𝑔 supp 𝑍) ≠ ∅)
3225, 31sylbi 207 . . . . . . . 8 (¬ (∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) → ∃𝑔𝑀 (𝑔 supp 𝑍) ≠ ∅)
3332ad2antrr 761 . . . . . . 7 (((¬ (∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉)) ∧ ∀𝑓𝑀 𝑓 finSupp 𝑍) → ∃𝑔𝑀 (𝑔 supp 𝑍) ≠ ∅)
34 iunn0 4571 . . . . . . 7 (∃𝑔𝑀 (𝑔 supp 𝑍) ≠ ∅ ↔ 𝑔𝑀 (𝑔 supp 𝑍) ≠ ∅)
3533, 34sylib 208 . . . . . 6 (((¬ (∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉)) ∧ ∀𝑓𝑀 𝑓 finSupp 𝑍) → 𝑔𝑀 (𝑔 supp 𝑍) ≠ ∅)
3618, 35jca 554 . . . . 5 (((¬ (∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉)) ∧ ∀𝑓𝑀 𝑓 finSupp 𝑍) → (∀𝑓𝑀 𝑓 finSupp 𝑍 𝑔𝑀 (𝑔 supp 𝑍) ≠ ∅))
37 oveq1 6642 . . . . . . 7 (𝑔 = 𝑓 → (𝑔 supp 𝑍) = (𝑓 supp 𝑍))
3837cbviunv 4550 . . . . . 6 𝑔𝑀 (𝑔 supp 𝑍) = 𝑓𝑀 (𝑓 supp 𝑍)
39 eqid 2620 . . . . . 6 sup( 𝑔𝑀 (𝑔 supp 𝑍), ℝ, < ) = sup( 𝑔𝑀 (𝑔 supp 𝑍), ℝ, < )
4038, 39fsuppmapnn0fiublem 12772 . . . . 5 ((𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉) → ((∀𝑓𝑀 𝑓 finSupp 𝑍 𝑔𝑀 (𝑔 supp 𝑍) ≠ ∅) → sup( 𝑔𝑀 (𝑔 supp 𝑍), ℝ, < ) ∈ ℕ0))
4117, 36, 40sylc 65 . . . 4 (((¬ (∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉)) ∧ ∀𝑓𝑀 𝑓 finSupp 𝑍) → sup( 𝑔𝑀 (𝑔 supp 𝑍), ℝ, < ) ∈ ℕ0)
42 nfv 1841 . . . . . . . . . 10 𝑓∅ = 𝑀
43 nfra1 2938 . . . . . . . . . 10 𝑓𝑓𝑀 (𝑓 supp 𝑍) = ∅
4442, 43nfor 1832 . . . . . . . . 9 𝑓(∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅)
4544nfn 1782 . . . . . . . 8 𝑓 ¬ (∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅)
46 nfv 1841 . . . . . . . 8 𝑓(𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉)
4745, 46nfan 1826 . . . . . . 7 𝑓(¬ (∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉))
48 nfra1 2938 . . . . . . 7 𝑓𝑓𝑀 𝑓 finSupp 𝑍
4947, 48nfan 1826 . . . . . 6 𝑓((¬ (∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉)) ∧ ∀𝑓𝑀 𝑓 finSupp 𝑍)
50 nfv 1841 . . . . . 6 𝑓 𝑚 = sup( 𝑔𝑀 (𝑔 supp 𝑍), ℝ, < )
5149, 50nfan 1826 . . . . 5 𝑓(((¬ (∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉)) ∧ ∀𝑓𝑀 𝑓 finSupp 𝑍) ∧ 𝑚 = sup( 𝑔𝑀 (𝑔 supp 𝑍), ℝ, < ))
52 oveq2 6643 . . . . . . 7 (𝑚 = sup( 𝑔𝑀 (𝑔 supp 𝑍), ℝ, < ) → (0...𝑚) = (0...sup( 𝑔𝑀 (𝑔 supp 𝑍), ℝ, < )))
5352sseq2d 3625 . . . . . 6 (𝑚 = sup( 𝑔𝑀 (𝑔 supp 𝑍), ℝ, < ) → ((𝑓 supp 𝑍) ⊆ (0...𝑚) ↔ (𝑓 supp 𝑍) ⊆ (0...sup( 𝑔𝑀 (𝑔 supp 𝑍), ℝ, < ))))
5453adantl 482 . . . . 5 ((((¬ (∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉)) ∧ ∀𝑓𝑀 𝑓 finSupp 𝑍) ∧ 𝑚 = sup( 𝑔𝑀 (𝑔 supp 𝑍), ℝ, < )) → ((𝑓 supp 𝑍) ⊆ (0...𝑚) ↔ (𝑓 supp 𝑍) ⊆ (0...sup( 𝑔𝑀 (𝑔 supp 𝑍), ℝ, < ))))
5551, 54ralbid 2980 . . . 4 ((((¬ (∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉)) ∧ ∀𝑓𝑀 𝑓 finSupp 𝑍) ∧ 𝑚 = sup( 𝑔𝑀 (𝑔 supp 𝑍), ℝ, < )) → (∀𝑓𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑚) ↔ ∀𝑓𝑀 (𝑓 supp 𝑍) ⊆ (0...sup( 𝑔𝑀 (𝑔 supp 𝑍), ℝ, < ))))
56 rexnal 2992 . . . . . . . . . . 11 (∃𝑓𝑀 ¬ (𝑓 supp 𝑍) = ∅ ↔ ¬ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅)
57 df-ne 2792 . . . . . . . . . . . . 13 ((𝑓 supp 𝑍) ≠ ∅ ↔ ¬ (𝑓 supp 𝑍) = ∅)
5857bicomi 214 . . . . . . . . . . . 12 (¬ (𝑓 supp 𝑍) = ∅ ↔ (𝑓 supp 𝑍) ≠ ∅)
5958rexbii 3037 . . . . . . . . . . 11 (∃𝑓𝑀 ¬ (𝑓 supp 𝑍) = ∅ ↔ ∃𝑓𝑀 (𝑓 supp 𝑍) ≠ ∅)
6056, 59sylbb1 227 . . . . . . . . . 10 (¬ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅ → ∃𝑓𝑀 (𝑓 supp 𝑍) ≠ ∅)
6160adantl 482 . . . . . . . . 9 ((¬ ∅ = 𝑀 ∧ ¬ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) → ∃𝑓𝑀 (𝑓 supp 𝑍) ≠ ∅)
6219, 61sylbi 207 . . . . . . . 8 (¬ (∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) → ∃𝑓𝑀 (𝑓 supp 𝑍) ≠ ∅)
6362ad2antrr 761 . . . . . . 7 (((¬ (∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉)) ∧ ∀𝑓𝑀 𝑓 finSupp 𝑍) → ∃𝑓𝑀 (𝑓 supp 𝑍) ≠ ∅)
64 iunn0 4571 . . . . . . . 8 (∃𝑓𝑀 (𝑓 supp 𝑍) ≠ ∅ ↔ 𝑓𝑀 (𝑓 supp 𝑍) ≠ ∅)
6520cbviunv 4550 . . . . . . . . 9 𝑓𝑀 (𝑓 supp 𝑍) = 𝑔𝑀 (𝑔 supp 𝑍)
6665neeq1i 2855 . . . . . . . 8 ( 𝑓𝑀 (𝑓 supp 𝑍) ≠ ∅ ↔ 𝑔𝑀 (𝑔 supp 𝑍) ≠ ∅)
6764, 66bitri 264 . . . . . . 7 (∃𝑓𝑀 (𝑓 supp 𝑍) ≠ ∅ ↔ 𝑔𝑀 (𝑔 supp 𝑍) ≠ ∅)
6863, 67sylib 208 . . . . . 6 (((¬ (∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉)) ∧ ∀𝑓𝑀 𝑓 finSupp 𝑍) → 𝑔𝑀 (𝑔 supp 𝑍) ≠ ∅)
6918, 68jca 554 . . . . 5 (((¬ (∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉)) ∧ ∀𝑓𝑀 𝑓 finSupp 𝑍) → (∀𝑓𝑀 𝑓 finSupp 𝑍 𝑔𝑀 (𝑔 supp 𝑍) ≠ ∅))
7038, 39fsuppmapnn0fiub 12773 . . . . 5 ((𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉) → ((∀𝑓𝑀 𝑓 finSupp 𝑍 𝑔𝑀 (𝑔 supp 𝑍) ≠ ∅) → ∀𝑓𝑀 (𝑓 supp 𝑍) ⊆ (0...sup( 𝑔𝑀 (𝑔 supp 𝑍), ℝ, < ))))
7117, 69, 70sylc 65 . . . 4 (((¬ (∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉)) ∧ ∀𝑓𝑀 𝑓 finSupp 𝑍) → ∀𝑓𝑀 (𝑓 supp 𝑍) ⊆ (0...sup( 𝑔𝑀 (𝑔 supp 𝑍), ℝ, < )))
7241, 55, 71rspcedvd 3312 . . 3 (((¬ (∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉)) ∧ ∀𝑓𝑀 𝑓 finSupp 𝑍) → ∃𝑚 ∈ ℕ0𝑓𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑚))
7372exp31 629 . 2 (¬ (∅ = 𝑀 ∨ ∀𝑓𝑀 (𝑓 supp 𝑍) = ∅) → ((𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉) → (∀𝑓𝑀 𝑓 finSupp 𝑍 → ∃𝑚 ∈ ℕ0𝑓𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑚))))
7416, 73pm2.61i 176 1 ((𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉) → (∀𝑓𝑀 𝑓 finSupp 𝑍 → ∃𝑚 ∈ ℕ0𝑓𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑚)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1481  wcel 1988  wne 2791  wral 2909  wrex 2910  wss 3567  c0 3907   ciun 4511   class class class wbr 4644  (class class class)co 6635   supp csupp 7280  𝑚 cmap 7842  Fincfn 7940   finSupp cfsupp 8260  supcsup 8331  cr 9920  0cc0 9921   < clt 10059  0cn0 11277  ...cfz 12311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-supp 7281  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-fsupp 8261  df-sup 8333  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-n0 11278  df-z 11363  df-uz 11673  df-fz 12312
This theorem is referenced by:  fsuppmapnn0fiub0  12776
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