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Theorem sticksstones2 40103
Description: The range function on strictly monotone functions with finite domain and codomain is an injective mapping onto 𝐾-elemental sets. (Contributed by metakunt, 27-Sep-2024.)
Hypotheses
Ref Expression
sticksstones2.1 (𝜑𝑁 ∈ ℕ0)
sticksstones2.2 (𝜑𝐾 ∈ ℕ0)
sticksstones2.3 𝐵 = {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾}
sticksstones2.4 𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))}
sticksstones2.5 𝐹 = (𝑧𝐴 ↦ ran 𝑧)
Assertion
Ref Expression
sticksstones2 (𝜑𝐹:𝐴1-1𝐵)
Distinct variable groups:   𝐴,𝑎,𝑧   𝐴,𝑓,𝑧   𝑧,𝐵   𝐾,𝑎,𝑥,𝑦   𝑓,𝐾,𝑥,𝑦   𝑁,𝑎   𝑓,𝑁   𝜑,𝑎,𝑧   𝜑,𝑓   𝑥,𝑧,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦,𝑓,𝑎)   𝐹(𝑥,𝑦,𝑧,𝑓,𝑎)   𝐾(𝑧)   𝑁(𝑥,𝑦,𝑧)

Proof of Theorem sticksstones2
Dummy variables 𝑏 𝑖 𝑗 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveqeq2 6783 . . . . . 6 (𝑎 = ran 𝑧 → ((♯‘𝑎) = 𝐾 ↔ (♯‘ran 𝑧) = 𝐾))
2 fzfid 13693 . . . . . . 7 ((𝜑𝑧𝐴) → (1...𝑁) ∈ Fin)
3 eleq1w 2821 . . . . . . . . . . . . 13 (𝑓 = 𝑧 → (𝑓𝐴𝑧𝐴))
4 feq1 6581 . . . . . . . . . . . . . 14 (𝑓 = 𝑧 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑧:(1...𝐾)⟶(1...𝑁)))
5 fveq1 6773 . . . . . . . . . . . . . . . . . 18 (𝑓 = 𝑧 → (𝑓𝑥) = (𝑧𝑥))
6 fveq1 6773 . . . . . . . . . . . . . . . . . 18 (𝑓 = 𝑧 → (𝑓𝑦) = (𝑧𝑦))
75, 6breq12d 5087 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝑧 → ((𝑓𝑥) < (𝑓𝑦) ↔ (𝑧𝑥) < (𝑧𝑦)))
87imbi2d 341 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑧 → ((𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)) ↔ (𝑥 < 𝑦 → (𝑧𝑥) < (𝑧𝑦))))
98ralbidv 3112 . . . . . . . . . . . . . . 15 (𝑓 = 𝑧 → (∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)) ↔ ∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧𝑥) < (𝑧𝑦))))
109ralbidv 3112 . . . . . . . . . . . . . 14 (𝑓 = 𝑧 → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)) ↔ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧𝑥) < (𝑧𝑦))))
114, 10anbi12d 631 . . . . . . . . . . . . 13 (𝑓 = 𝑧 → ((𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦))) ↔ (𝑧:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧𝑥) < (𝑧𝑦)))))
123, 11bibi12d 346 . . . . . . . . . . . 12 (𝑓 = 𝑧 → ((𝑓𝐴 ↔ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))) ↔ (𝑧𝐴 ↔ (𝑧:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧𝑥) < (𝑧𝑦))))))
13 sticksstones2.4 . . . . . . . . . . . . . 14 𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))}
14 abeq2 2872 . . . . . . . . . . . . . 14 (𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))} ↔ ∀𝑓(𝑓𝐴 ↔ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))))
1513, 14mpbi 229 . . . . . . . . . . . . 13 𝑓(𝑓𝐴 ↔ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦))))
1615spi 2177 . . . . . . . . . . . 12 (𝑓𝐴 ↔ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦))))
1712, 16chvarvv 2002 . . . . . . . . . . 11 (𝑧𝐴 ↔ (𝑧:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧𝑥) < (𝑧𝑦))))
1817biimpi 215 . . . . . . . . . 10 (𝑧𝐴 → (𝑧:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧𝑥) < (𝑧𝑦))))
1918adantl 482 . . . . . . . . 9 ((𝜑𝑧𝐴) → (𝑧:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧𝑥) < (𝑧𝑦))))
2019simpld 495 . . . . . . . 8 ((𝜑𝑧𝐴) → 𝑧:(1...𝐾)⟶(1...𝑁))
2120frnd 6608 . . . . . . 7 ((𝜑𝑧𝐴) → ran 𝑧 ⊆ (1...𝑁))
222, 21sselpwd 5250 . . . . . 6 ((𝜑𝑧𝐴) → ran 𝑧 ∈ 𝒫 (1...𝑁))
2320ffnd 6601 . . . . . . . . . . 11 ((𝜑𝑧𝐴) → 𝑧 Fn (1...𝐾))
24 hashfn 14090 . . . . . . . . . . 11 (𝑧 Fn (1...𝐾) → (♯‘𝑧) = (♯‘(1...𝐾)))
2523, 24syl 17 . . . . . . . . . 10 ((𝜑𝑧𝐴) → (♯‘𝑧) = (♯‘(1...𝐾)))
26 sticksstones2.2 . . . . . . . . . . . 12 (𝜑𝐾 ∈ ℕ0)
2726adantr 481 . . . . . . . . . . 11 ((𝜑𝑧𝐴) → 𝐾 ∈ ℕ0)
28 hashfz1 14060 . . . . . . . . . . 11 (𝐾 ∈ ℕ0 → (♯‘(1...𝐾)) = 𝐾)
2927, 28syl 17 . . . . . . . . . 10 ((𝜑𝑧𝐴) → (♯‘(1...𝐾)) = 𝐾)
3025, 29eqtrd 2778 . . . . . . . . 9 ((𝜑𝑧𝐴) → (♯‘𝑧) = 𝐾)
3130eqcomd 2744 . . . . . . . 8 ((𝜑𝑧𝐴) → 𝐾 = (♯‘𝑧))
32 fzfid 13693 . . . . . . . . 9 ((𝜑𝑧𝐴) → (1...𝐾) ∈ Fin)
33 elfznn 13285 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ (1...𝐾) → 𝑎 ∈ ℕ)
34333ad2ant3 1134 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) → 𝑎 ∈ ℕ)
3534nnred 11988 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) → 𝑎 ∈ ℝ)
3635adantr 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → 𝑎 ∈ ℝ)
37 elfznn 13285 . . . . . . . . . . . . . . . . . . 19 (𝑏 ∈ (1...𝐾) → 𝑏 ∈ ℕ)
3837nnred 11988 . . . . . . . . . . . . . . . . . 18 (𝑏 ∈ (1...𝐾) → 𝑏 ∈ ℝ)
3938adantl 482 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → 𝑏 ∈ ℝ)
40 lttri2 11057 . . . . . . . . . . . . . . . . 17 ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎𝑏 ↔ (𝑎 < 𝑏𝑏 < 𝑎)))
4136, 39, 40syl2anc 584 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑎𝑏 ↔ (𝑎 < 𝑏𝑏 < 𝑎)))
42203adant3 1131 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) → 𝑧:(1...𝐾)⟶(1...𝑁))
43 simp3 1137 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) → 𝑎 ∈ (1...𝐾))
4442, 43ffvelrnd 6962 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) → (𝑧𝑎) ∈ (1...𝑁))
4544adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑧𝑎) ∈ (1...𝑁))
4645adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑎 < 𝑏) → (𝑧𝑎) ∈ (1...𝑁))
47 elfznn 13285 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧𝑎) ∈ (1...𝑁) → (𝑧𝑎) ∈ ℕ)
4846, 47syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑎 < 𝑏) → (𝑧𝑎) ∈ ℕ)
4948nnred 11988 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑎 < 𝑏) → (𝑧𝑎) ∈ ℝ)
5019simprd 496 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧𝐴) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧𝑥) < (𝑧𝑦)))
51503adant3 1131 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧𝑥) < (𝑧𝑦)))
5251adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧𝑥) < (𝑧𝑦)))
5343adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → 𝑎 ∈ (1...𝐾))
54 simpr 485 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → 𝑏 ∈ (1...𝐾))
55 breq1 5077 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑎 → (𝑥 < 𝑦𝑎 < 𝑦))
56 fveq2 6774 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑎 → (𝑧𝑥) = (𝑧𝑎))
5756breq1d 5084 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑎 → ((𝑧𝑥) < (𝑧𝑦) ↔ (𝑧𝑎) < (𝑧𝑦)))
5855, 57imbi12d 345 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑎 → ((𝑥 < 𝑦 → (𝑧𝑥) < (𝑧𝑦)) ↔ (𝑎 < 𝑦 → (𝑧𝑎) < (𝑧𝑦))))
59 breq2 5078 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑏 → (𝑎 < 𝑦𝑎 < 𝑏))
60 fveq2 6774 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑏 → (𝑧𝑦) = (𝑧𝑏))
6160breq2d 5086 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑏 → ((𝑧𝑎) < (𝑧𝑦) ↔ (𝑧𝑎) < (𝑧𝑏)))
6259, 61imbi12d 345 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = 𝑏 → ((𝑎 < 𝑦 → (𝑧𝑎) < (𝑧𝑦)) ↔ (𝑎 < 𝑏 → (𝑧𝑎) < (𝑧𝑏))))
6358, 62rspc2v 3570 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 ∈ (1...𝐾) ∧ 𝑏 ∈ (1...𝐾)) → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧𝑥) < (𝑧𝑦)) → (𝑎 < 𝑏 → (𝑧𝑎) < (𝑧𝑏))))
6453, 54, 63syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧𝑥) < (𝑧𝑦)) → (𝑎 < 𝑏 → (𝑧𝑎) < (𝑧𝑏))))
6552, 64mpd 15 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑎 < 𝑏 → (𝑧𝑎) < (𝑧𝑏)))
6665imp 407 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑎 < 𝑏) → (𝑧𝑎) < (𝑧𝑏))
6749, 66ltned 11111 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑎 < 𝑏) → (𝑧𝑎) ≠ (𝑧𝑏))
6842ffvelrnda 6961 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑧𝑏) ∈ (1...𝑁))
69 elfznn 13285 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑧𝑏) ∈ (1...𝑁) → (𝑧𝑏) ∈ ℕ)
7068, 69syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑧𝑏) ∈ ℕ)
7170nnred 11988 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑧𝑏) ∈ ℝ)
7271adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑏 < 𝑎) → (𝑧𝑏) ∈ ℝ)
73 breq1 5077 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑏 → (𝑥 < 𝑦𝑏 < 𝑦))
74 fveq2 6774 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝑏 → (𝑧𝑥) = (𝑧𝑏))
7574breq1d 5084 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑏 → ((𝑧𝑥) < (𝑧𝑦) ↔ (𝑧𝑏) < (𝑧𝑦)))
7673, 75imbi12d 345 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑏 → ((𝑥 < 𝑦 → (𝑧𝑥) < (𝑧𝑦)) ↔ (𝑏 < 𝑦 → (𝑧𝑏) < (𝑧𝑦))))
77 breq2 5078 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑎 → (𝑏 < 𝑦𝑏 < 𝑎))
78 fveq2 6774 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑎 → (𝑧𝑦) = (𝑧𝑎))
7978breq2d 5086 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑎 → ((𝑧𝑏) < (𝑧𝑦) ↔ (𝑧𝑏) < (𝑧𝑎)))
8077, 79imbi12d 345 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑎 → ((𝑏 < 𝑦 → (𝑧𝑏) < (𝑧𝑦)) ↔ (𝑏 < 𝑎 → (𝑧𝑏) < (𝑧𝑎))))
8176, 80rspc2v 3570 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏 ∈ (1...𝐾) ∧ 𝑎 ∈ (1...𝐾)) → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧𝑥) < (𝑧𝑦)) → (𝑏 < 𝑎 → (𝑧𝑏) < (𝑧𝑎))))
8254, 53, 81syl2anc 584 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧𝑥) < (𝑧𝑦)) → (𝑏 < 𝑎 → (𝑧𝑏) < (𝑧𝑎))))
8352, 82mpd 15 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑏 < 𝑎 → (𝑧𝑏) < (𝑧𝑎)))
8483imp 407 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑏 < 𝑎) → (𝑧𝑏) < (𝑧𝑎))
8572, 84ltned 11111 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑏 < 𝑎) → (𝑧𝑏) ≠ (𝑧𝑎))
8685necomd 2999 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑏 < 𝑎) → (𝑧𝑎) ≠ (𝑧𝑏))
8767, 86jaodan 955 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ (𝑎 < 𝑏𝑏 < 𝑎)) → (𝑧𝑎) ≠ (𝑧𝑏))
8887ex 413 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → ((𝑎 < 𝑏𝑏 < 𝑎) → (𝑧𝑎) ≠ (𝑧𝑏)))
8941, 88sylbid 239 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑎𝑏 → (𝑧𝑎) ≠ (𝑧𝑏)))
9089necon4d 2967 . . . . . . . . . . . . . 14 (((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → ((𝑧𝑎) = (𝑧𝑏) → 𝑎 = 𝑏))
9190ralrimiva 3103 . . . . . . . . . . . . 13 ((𝜑𝑧𝐴𝑎 ∈ (1...𝐾)) → ∀𝑏 ∈ (1...𝐾)((𝑧𝑎) = (𝑧𝑏) → 𝑎 = 𝑏))
92913expa 1117 . . . . . . . . . . . 12 (((𝜑𝑧𝐴) ∧ 𝑎 ∈ (1...𝐾)) → ∀𝑏 ∈ (1...𝐾)((𝑧𝑎) = (𝑧𝑏) → 𝑎 = 𝑏))
9392ralrimiva 3103 . . . . . . . . . . 11 ((𝜑𝑧𝐴) → ∀𝑎 ∈ (1...𝐾)∀𝑏 ∈ (1...𝐾)((𝑧𝑎) = (𝑧𝑏) → 𝑎 = 𝑏))
9420, 93jca 512 . . . . . . . . . 10 ((𝜑𝑧𝐴) → (𝑧:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑎 ∈ (1...𝐾)∀𝑏 ∈ (1...𝐾)((𝑧𝑎) = (𝑧𝑏) → 𝑎 = 𝑏)))
95 dff13 7128 . . . . . . . . . 10 (𝑧:(1...𝐾)–1-1→(1...𝑁) ↔ (𝑧:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑎 ∈ (1...𝐾)∀𝑏 ∈ (1...𝐾)((𝑧𝑎) = (𝑧𝑏) → 𝑎 = 𝑏)))
9694, 95sylibr 233 . . . . . . . . 9 ((𝜑𝑧𝐴) → 𝑧:(1...𝐾)–1-1→(1...𝑁))
97 hashf1rn 14067 . . . . . . . . 9 (((1...𝐾) ∈ Fin ∧ 𝑧:(1...𝐾)–1-1→(1...𝑁)) → (♯‘𝑧) = (♯‘ran 𝑧))
9832, 96, 97syl2anc 584 . . . . . . . 8 ((𝜑𝑧𝐴) → (♯‘𝑧) = (♯‘ran 𝑧))
9931, 98eqtrd 2778 . . . . . . 7 ((𝜑𝑧𝐴) → 𝐾 = (♯‘ran 𝑧))
10099eqcomd 2744 . . . . . 6 ((𝜑𝑧𝐴) → (♯‘ran 𝑧) = 𝐾)
1011, 22, 100elrabd 3626 . . . . 5 ((𝜑𝑧𝐴) → ran 𝑧 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾})
102 sticksstones2.3 . . . . . . 7 𝐵 = {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾}
103102eleq2i 2830 . . . . . 6 (ran 𝑧𝐵 ↔ ran 𝑧 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾})
104103a1i 11 . . . . 5 ((𝜑𝑧𝐴) → (ran 𝑧𝐵 ↔ ran 𝑧 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾}))
105101, 104mpbird 256 . . . 4 ((𝜑𝑧𝐴) → ran 𝑧𝐵)
106 sticksstones2.5 . . . 4 𝐹 = (𝑧𝐴 ↦ ran 𝑧)
107105, 106fmptd 6988 . . 3 (𝜑𝐹:𝐴𝐵)
108 sticksstones2.1 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ0)
1091083ad2ant1 1132 . . . . . . . . . . 11 ((𝜑𝑖𝐴𝑗𝐴) → 𝑁 ∈ ℕ0)
110109adantr 481 . . . . . . . . . 10 (((𝜑𝑖𝐴𝑗𝐴) ∧ 𝑖𝑗) → 𝑁 ∈ ℕ0)
111263ad2ant1 1132 . . . . . . . . . . 11 ((𝜑𝑖𝐴𝑗𝐴) → 𝐾 ∈ ℕ0)
112111adantr 481 . . . . . . . . . 10 (((𝜑𝑖𝐴𝑗𝐴) ∧ 𝑖𝑗) → 𝐾 ∈ ℕ0)
113 simpl2 1191 . . . . . . . . . 10 (((𝜑𝑖𝐴𝑗𝐴) ∧ 𝑖𝑗) → 𝑖𝐴)
114 simpl3 1192 . . . . . . . . . 10 (((𝜑𝑖𝐴𝑗𝐴) ∧ 𝑖𝑗) → 𝑗𝐴)
115 simpr 485 . . . . . . . . . 10 (((𝜑𝑖𝐴𝑗𝐴) ∧ 𝑖𝑗) → 𝑖𝑗)
116 fveq2 6774 . . . . . . . . . . . . 13 (𝑟 = 𝑠 → (𝑖𝑟) = (𝑖𝑠))
117 fveq2 6774 . . . . . . . . . . . . 13 (𝑟 = 𝑠 → (𝑗𝑟) = (𝑗𝑠))
118116, 117neeq12d 3005 . . . . . . . . . . . 12 (𝑟 = 𝑠 → ((𝑖𝑟) ≠ (𝑗𝑟) ↔ (𝑖𝑠) ≠ (𝑗𝑠)))
119118cbvrabv 3426 . . . . . . . . . . 11 {𝑟 ∈ (1...𝐾) ∣ (𝑖𝑟) ≠ (𝑗𝑟)} = {𝑠 ∈ (1...𝐾) ∣ (𝑖𝑠) ≠ (𝑗𝑠)}
120119infeq1i 9237 . . . . . . . . . 10 inf({𝑟 ∈ (1...𝐾) ∣ (𝑖𝑟) ≠ (𝑗𝑟)}, ℝ, < ) = inf({𝑠 ∈ (1...𝐾) ∣ (𝑖𝑠) ≠ (𝑗𝑠)}, ℝ, < )
121110, 112, 13, 113, 114, 115, 120sticksstones1 40102 . . . . . . . . 9 (((𝜑𝑖𝐴𝑗𝐴) ∧ 𝑖𝑗) → ran 𝑖 ≠ ran 𝑗)
122106a1i 11 . . . . . . . . . 10 (((𝜑𝑖𝐴𝑗𝐴) ∧ 𝑖𝑗) → 𝐹 = (𝑧𝐴 ↦ ran 𝑧))
123 simpr 485 . . . . . . . . . . 11 ((((𝜑𝑖𝐴𝑗𝐴) ∧ 𝑖𝑗) ∧ 𝑧 = 𝑖) → 𝑧 = 𝑖)
124123rneqd 5847 . . . . . . . . . 10 ((((𝜑𝑖𝐴𝑗𝐴) ∧ 𝑖𝑗) ∧ 𝑧 = 𝑖) → ran 𝑧 = ran 𝑖)
125 fzfid 13693 . . . . . . . . . . 11 (((𝜑𝑖𝐴𝑗𝐴) ∧ 𝑖𝑗) → (1...𝑁) ∈ Fin)
126 eleq1w 2821 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝑖 → (𝑓𝐴𝑖𝐴))
127 feq1 6581 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝑖 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑖:(1...𝐾)⟶(1...𝑁)))
128 fveq1 6773 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 = 𝑖 → (𝑓𝑥) = (𝑖𝑥))
129 fveq1 6773 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 = 𝑖 → (𝑓𝑦) = (𝑖𝑦))
130128, 129breq12d 5087 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = 𝑖 → ((𝑓𝑥) < (𝑓𝑦) ↔ (𝑖𝑥) < (𝑖𝑦)))
131130imbi2d 341 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝑖 → ((𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)) ↔ (𝑥 < 𝑦 → (𝑖𝑥) < (𝑖𝑦))))
1321312ralbidv 3129 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝑖 → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)) ↔ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑖𝑥) < (𝑖𝑦))))
133127, 132anbi12d 631 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝑖 → ((𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦))) ↔ (𝑖:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑖𝑥) < (𝑖𝑦)))))
134126, 133bibi12d 346 . . . . . . . . . . . . . . . . . 18 (𝑓 = 𝑖 → ((𝑓𝐴 ↔ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))) ↔ (𝑖𝐴 ↔ (𝑖:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑖𝑥) < (𝑖𝑦))))))
135134, 16chvarvv 2002 . . . . . . . . . . . . . . . . 17 (𝑖𝐴 ↔ (𝑖:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑖𝑥) < (𝑖𝑦))))
136135biimpi 215 . . . . . . . . . . . . . . . 16 (𝑖𝐴 → (𝑖:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑖𝑥) < (𝑖𝑦))))
137136adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝐴) → (𝑖:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑖𝑥) < (𝑖𝑦))))
138137simpld 495 . . . . . . . . . . . . . 14 ((𝜑𝑖𝐴) → 𝑖:(1...𝐾)⟶(1...𝑁))
1391383adant3 1131 . . . . . . . . . . . . 13 ((𝜑𝑖𝐴𝑗𝐴) → 𝑖:(1...𝐾)⟶(1...𝑁))
140139adantr 481 . . . . . . . . . . . 12 (((𝜑𝑖𝐴𝑗𝐴) ∧ 𝑖𝑗) → 𝑖:(1...𝐾)⟶(1...𝑁))
141140frnd 6608 . . . . . . . . . . 11 (((𝜑𝑖𝐴𝑗𝐴) ∧ 𝑖𝑗) → ran 𝑖 ⊆ (1...𝑁))
142125, 141sselpwd 5250 . . . . . . . . . 10 (((𝜑𝑖𝐴𝑗𝐴) ∧ 𝑖𝑗) → ran 𝑖 ∈ 𝒫 (1...𝑁))
143122, 124, 113, 142fvmptd 6882 . . . . . . . . 9 (((𝜑𝑖𝐴𝑗𝐴) ∧ 𝑖𝑗) → (𝐹𝑖) = ran 𝑖)
144 simpr 485 . . . . . . . . . . 11 ((((𝜑𝑖𝐴𝑗𝐴) ∧ 𝑖𝑗) ∧ 𝑧 = 𝑗) → 𝑧 = 𝑗)
145144rneqd 5847 . . . . . . . . . 10 ((((𝜑𝑖𝐴𝑗𝐴) ∧ 𝑖𝑗) ∧ 𝑧 = 𝑗) → ran 𝑧 = ran 𝑗)
146 fzfid 13693 . . . . . . . . . . . . 13 (𝜑 → (1...𝑁) ∈ Fin)
1471463ad2ant1 1132 . . . . . . . . . . . 12 ((𝜑𝑖𝐴𝑗𝐴) → (1...𝑁) ∈ Fin)
148 eleq1w 2821 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝑗 → (𝑓𝐴𝑗𝐴))
149 feq1 6581 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝑗 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑗:(1...𝐾)⟶(1...𝑁)))
150 fveq1 6773 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 = 𝑗 → (𝑓𝑥) = (𝑗𝑥))
151 fveq1 6773 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 = 𝑗 → (𝑓𝑦) = (𝑗𝑦))
152150, 151breq12d 5087 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = 𝑗 → ((𝑓𝑥) < (𝑓𝑦) ↔ (𝑗𝑥) < (𝑗𝑦)))
153152imbi2d 341 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝑗 → ((𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)) ↔ (𝑥 < 𝑦 → (𝑗𝑥) < (𝑗𝑦))))
1541532ralbidv 3129 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝑗 → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)) ↔ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑗𝑥) < (𝑗𝑦))))
155149, 154anbi12d 631 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝑗 → ((𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦))) ↔ (𝑗:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑗𝑥) < (𝑗𝑦)))))
156148, 155bibi12d 346 . . . . . . . . . . . . . . . . . 18 (𝑓 = 𝑗 → ((𝑓𝐴 ↔ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))) ↔ (𝑗𝐴 ↔ (𝑗:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑗𝑥) < (𝑗𝑦))))))
157156, 16chvarvv 2002 . . . . . . . . . . . . . . . . 17 (𝑗𝐴 ↔ (𝑗:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑗𝑥) < (𝑗𝑦))))
158157biimpi 215 . . . . . . . . . . . . . . . 16 (𝑗𝐴 → (𝑗:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑗𝑥) < (𝑗𝑦))))
159158adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑗𝐴) → (𝑗:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑗𝑥) < (𝑗𝑦))))
160159simpld 495 . . . . . . . . . . . . . 14 ((𝜑𝑗𝐴) → 𝑗:(1...𝐾)⟶(1...𝑁))
1611603adant2 1130 . . . . . . . . . . . . 13 ((𝜑𝑖𝐴𝑗𝐴) → 𝑗:(1...𝐾)⟶(1...𝑁))
162161frnd 6608 . . . . . . . . . . . 12 ((𝜑𝑖𝐴𝑗𝐴) → ran 𝑗 ⊆ (1...𝑁))
163147, 162sselpwd 5250 . . . . . . . . . . 11 ((𝜑𝑖𝐴𝑗𝐴) → ran 𝑗 ∈ 𝒫 (1...𝑁))
164163adantr 481 . . . . . . . . . 10 (((𝜑𝑖𝐴𝑗𝐴) ∧ 𝑖𝑗) → ran 𝑗 ∈ 𝒫 (1...𝑁))
165122, 145, 114, 164fvmptd 6882 . . . . . . . . 9 (((𝜑𝑖𝐴𝑗𝐴) ∧ 𝑖𝑗) → (𝐹𝑗) = ran 𝑗)
166121, 143, 1653netr4d 3021 . . . . . . . 8 (((𝜑𝑖𝐴𝑗𝐴) ∧ 𝑖𝑗) → (𝐹𝑖) ≠ (𝐹𝑗))
167166ex 413 . . . . . . 7 ((𝜑𝑖𝐴𝑗𝐴) → (𝑖𝑗 → (𝐹𝑖) ≠ (𝐹𝑗)))
168167necon4d 2967 . . . . . 6 ((𝜑𝑖𝐴𝑗𝐴) → ((𝐹𝑖) = (𝐹𝑗) → 𝑖 = 𝑗))
1691683expa 1117 . . . . 5 (((𝜑𝑖𝐴) ∧ 𝑗𝐴) → ((𝐹𝑖) = (𝐹𝑗) → 𝑖 = 𝑗))
170169ralrimiva 3103 . . . 4 ((𝜑𝑖𝐴) → ∀𝑗𝐴 ((𝐹𝑖) = (𝐹𝑗) → 𝑖 = 𝑗))
171170ralrimiva 3103 . . 3 (𝜑 → ∀𝑖𝐴𝑗𝐴 ((𝐹𝑖) = (𝐹𝑗) → 𝑖 = 𝑗))
172107, 171jca 512 . 2 (𝜑 → (𝐹:𝐴𝐵 ∧ ∀𝑖𝐴𝑗𝐴 ((𝐹𝑖) = (𝐹𝑗) → 𝑖 = 𝑗)))
173 dff13 7128 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑖𝐴𝑗𝐴 ((𝐹𝑖) = (𝐹𝑗) → 𝑖 = 𝑗)))
174172, 173sylibr 233 1 (𝜑𝐹:𝐴1-1𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 844  w3a 1086  wal 1537   = wceq 1539  wcel 2106  {cab 2715  wne 2943  wral 3064  {crab 3068  𝒫 cpw 4533   class class class wbr 5074  cmpt 5157  ran crn 5590   Fn wfn 6428  wf 6429  1-1wf1 6430  cfv 6433  (class class class)co 7275  Fincfn 8733  infcinf 9200  cr 10870  1c1 10872   < clt 11009  cn 11973  0cn0 12233  ...cfz 13239  chash 14044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-inf 9202  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-hash 14045
This theorem is referenced by:  sticksstones3  40104  sticksstones4  40105
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