sticksstones2 - Mathbox for metakunt

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Theorem sticksstones2 39772

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	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
sticksstones2.2	⊢ (𝜑 → 𝐾 ∈ 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 → (𝑓‘𝑥) < (𝑓‘𝑦)))}
sticksstones2.5	⊢ 𝐹 = (𝑧 ∈ 𝐴 ↦ ran 𝑧)

**Assertion**
Ref	Expression
sticksstones2	⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)

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.

Step	Hyp	Ref	Expression
1		fveqeq2 6704	. . . . . 6 ⊢ (𝑎 = ran 𝑧 → ((♯‘𝑎) = 𝐾 ↔ (♯‘ran 𝑧) = 𝐾))
2		fzfid 13511	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (1...𝑁) ∈ Fin)
3		eleq1w 2813	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑧 → (𝑓 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
4		feq1 6504	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑧 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑧:(1...𝐾)⟶(1...𝑁)))
5		fveq1 6694	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = 𝑧 → (𝑓‘𝑥) = (𝑧‘𝑥))
6		fveq1 6694	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = 𝑧 → (𝑓‘𝑦) = (𝑧‘𝑦))
7	5, 6	breq12d 5052	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝑧 → ((𝑓‘𝑥) < (𝑓‘𝑦) ↔ (𝑧‘𝑥) < (𝑧‘𝑦)))
8	7	imbi2d 344	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑧 → ((𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ (𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦))))
9	8	ralbidv 3108	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑧 → (∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ ∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦))))
10	9	ralbidv 3108	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑧 → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦))))
11	4, 10	anbi12d 634	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑧 → ((𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦))) ↔ (𝑧:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦)))))
12	3, 11	bibi12d 349	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑧 → ((𝑓 ∈ 𝐴 ↔ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))) ↔ (𝑧 ∈ 𝐴 ↔ (𝑧:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦))))))
13		sticksstones2.4	. . . . . . . . . . . . . 14 ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}
14		abeq2 2862	. . . . . . . . . . . . . 14 ⊢ (𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))} ↔ ∀𝑓(𝑓 ∈ 𝐴 ↔ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))))
15	13, 14	mpbi 233	. . . . . . . . . . . . 13 ⊢ ∀𝑓(𝑓 ∈ 𝐴 ↔ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦))))
16	15	spi 2183	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝐴 ↔ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦))))
17	12, 16	chvarvv 2008	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐴 ↔ (𝑧:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦))))
18	17	biimpi 219	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐴 → (𝑧:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦))))
19	18	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝑧:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦))))
20	19	simpld 498	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧:(1...𝐾)⟶(1...𝑁))
21	20	frnd 6531	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ran 𝑧 ⊆ (1...𝑁))
22	2, 21	sselpwd 5204	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ran 𝑧 ∈ 𝒫 (1...𝑁))
23	20	ffnd 6524	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 Fn (1...𝐾))
24		hashfn 13907	. . . . . . . . . . 11 ⊢ (𝑧 Fn (1...𝐾) → (♯‘𝑧) = (♯‘(1...𝐾)))
25	23, 24	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (♯‘𝑧) = (♯‘(1...𝐾)))
26		sticksstones2.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
27	26	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐾 ∈ ℕ₀)
28		hashfz1 13877	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℕ₀ → (♯‘(1...𝐾)) = 𝐾)
29	27, 28	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (♯‘(1...𝐾)) = 𝐾)
30	25, 29	eqtrd 2771	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (♯‘𝑧) = 𝐾)
31	30	eqcomd 2742	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐾 = (♯‘𝑧))
32		fzfid 13511	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (1...𝐾) ∈ Fin)
33		elfznn 13106	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ (1...𝐾) → 𝑎 ∈ ℕ)
34	33	3ad2ant3 1137	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) → 𝑎 ∈ ℕ)
35	34	nnred 11810	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) → 𝑎 ∈ ℝ)
36	35	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → 𝑎 ∈ ℝ)
37		elfznn 13106	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∈ (1...𝐾) → 𝑏 ∈ ℕ)
38	37	nnred 11810	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ (1...𝐾) → 𝑏 ∈ ℝ)
39	38	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → 𝑏 ∈ ℝ)
40		lttri2 10880	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎 ≠ 𝑏 ↔ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎)))
41	36, 39, 40	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑎 ≠ 𝑏 ↔ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎)))
42	20	3adant3 1134	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) → 𝑧:(1...𝐾)⟶(1...𝑁))
43		simp3 1140	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) → 𝑎 ∈ (1...𝐾))
44	42, 43	ffvelrnd 6883	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) → (𝑧‘𝑎) ∈ (1...𝑁))
45	44	adantr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑧‘𝑎) ∈ (1...𝑁))
46	45	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑎 < 𝑏) → (𝑧‘𝑎) ∈ (1...𝑁))
47		elfznn 13106	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧‘𝑎) ∈ (1...𝑁) → (𝑧‘𝑎) ∈ ℕ)
48	46, 47	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑎 < 𝑏) → (𝑧‘𝑎) ∈ ℕ)
49	48	nnred 11810	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑎 < 𝑏) → (𝑧‘𝑎) ∈ ℝ)
50	19	simprd 499	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦)))
51	50	3adant3 1134	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦)))
52	51	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦)))
53	43	adantr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → 𝑎 ∈ (1...𝐾))
54		simpr 488	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → 𝑏 ∈ (1...𝐾))
55		breq1 5042	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑎 → (𝑥 < 𝑦 ↔ 𝑎 < 𝑦))
56		fveq2 6695	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = 𝑎 → (𝑧‘𝑥) = (𝑧‘𝑎))
57	56	breq1d 5049	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑎 → ((𝑧‘𝑥) < (𝑧‘𝑦) ↔ (𝑧‘𝑎) < (𝑧‘𝑦)))
58	55, 57	imbi12d 348	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑎 → ((𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦)) ↔ (𝑎 < 𝑦 → (𝑧‘𝑎) < (𝑧‘𝑦))))
59		breq2 5043	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑏 → (𝑎 < 𝑦 ↔ 𝑎 < 𝑏))
60		fveq2 6695	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = 𝑏 → (𝑧‘𝑦) = (𝑧‘𝑏))
61	60	breq2d 5051	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑏 → ((𝑧‘𝑎) < (𝑧‘𝑦) ↔ (𝑧‘𝑎) < (𝑧‘𝑏)))
62	59, 61	imbi12d 348	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑏 → ((𝑎 < 𝑦 → (𝑧‘𝑎) < (𝑧‘𝑦)) ↔ (𝑎 < 𝑏 → (𝑧‘𝑎) < (𝑧‘𝑏))))
63	58, 62	rspc2v 3537	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 ∈ (1...𝐾) ∧ 𝑏 ∈ (1...𝐾)) → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦)) → (𝑎 < 𝑏 → (𝑧‘𝑎) < (𝑧‘𝑏))))
64	53, 54, 63	syl2anc 587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦)) → (𝑎 < 𝑏 → (𝑧‘𝑎) < (𝑧‘𝑏))))
65	52, 64	mpd 15	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑎 < 𝑏 → (𝑧‘𝑎) < (𝑧‘𝑏)))
66	65	imp 410	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑎 < 𝑏) → (𝑧‘𝑎) < (𝑧‘𝑏))
67	49, 66	ltned 10933	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑎 < 𝑏) → (𝑧‘𝑎) ≠ (𝑧‘𝑏))
68	42	ffvelrnda 6882	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑧‘𝑏) ∈ (1...𝑁))
69		elfznn 13106	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧‘𝑏) ∈ (1...𝑁) → (𝑧‘𝑏) ∈ ℕ)
70	68, 69	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑧‘𝑏) ∈ ℕ)
71	70	nnred 11810	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑧‘𝑏) ∈ ℝ)
72	71	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑏 < 𝑎) → (𝑧‘𝑏) ∈ ℝ)
73		breq1 5042	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = 𝑏 → (𝑥 < 𝑦 ↔ 𝑏 < 𝑦))
74		fveq2 6695	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = 𝑏 → (𝑧‘𝑥) = (𝑧‘𝑏))
75	74	breq1d 5049	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = 𝑏 → ((𝑧‘𝑥) < (𝑧‘𝑦) ↔ (𝑧‘𝑏) < (𝑧‘𝑦)))
76	73, 75	imbi12d 348	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑏 → ((𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦)) ↔ (𝑏 < 𝑦 → (𝑧‘𝑏) < (𝑧‘𝑦))))
77		breq2 5043	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = 𝑎 → (𝑏 < 𝑦 ↔ 𝑏 < 𝑎))
78		fveq2 6695	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 = 𝑎 → (𝑧‘𝑦) = (𝑧‘𝑎))
79	78	breq2d 5051	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = 𝑎 → ((𝑧‘𝑏) < (𝑧‘𝑦) ↔ (𝑧‘𝑏) < (𝑧‘𝑎)))
80	77, 79	imbi12d 348	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑎 → ((𝑏 < 𝑦 → (𝑧‘𝑏) < (𝑧‘𝑦)) ↔ (𝑏 < 𝑎 → (𝑧‘𝑏) < (𝑧‘𝑎))))
81	76, 80	rspc2v 3537	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑏 ∈ (1...𝐾) ∧ 𝑎 ∈ (1...𝐾)) → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦)) → (𝑏 < 𝑎 → (𝑧‘𝑏) < (𝑧‘𝑎))))
82	54, 53, 81	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦)) → (𝑏 < 𝑎 → (𝑧‘𝑏) < (𝑧‘𝑎))))
83	52, 82	mpd 15	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑏 < 𝑎 → (𝑧‘𝑏) < (𝑧‘𝑎)))
84	83	imp 410	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑏 < 𝑎) → (𝑧‘𝑏) < (𝑧‘𝑎))
85	72, 84	ltned 10933	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑏 < 𝑎) → (𝑧‘𝑏) ≠ (𝑧‘𝑎))
86	85	necomd 2987	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑏 < 𝑎) → (𝑧‘𝑎) ≠ (𝑧‘𝑏))
87	67, 86	jaodan 958	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎)) → (𝑧‘𝑎) ≠ (𝑧‘𝑏))
88	87	ex 416	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → ((𝑎 < 𝑏 ∨ 𝑏 < 𝑎) → (𝑧‘𝑎) ≠ (𝑧‘𝑏)))
89	41, 88	sylbid 243	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑎 ≠ 𝑏 → (𝑧‘𝑎) ≠ (𝑧‘𝑏)))
90	89	necon4d 2956	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → ((𝑧‘𝑎) = (𝑧‘𝑏) → 𝑎 = 𝑏))
91	90	ralrimiva 3095	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) → ∀𝑏 ∈ (1...𝐾)((𝑧‘𝑎) = (𝑧‘𝑏) → 𝑎 = 𝑏))
92	91	3expa 1120	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑎 ∈ (1...𝐾)) → ∀𝑏 ∈ (1...𝐾)((𝑧‘𝑎) = (𝑧‘𝑏) → 𝑎 = 𝑏))
93	92	ralrimiva 3095	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ∀𝑎 ∈ (1...𝐾)∀𝑏 ∈ (1...𝐾)((𝑧‘𝑎) = (𝑧‘𝑏) → 𝑎 = 𝑏))
94	20, 93	jca 515	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝑧:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑎 ∈ (1...𝐾)∀𝑏 ∈ (1...𝐾)((𝑧‘𝑎) = (𝑧‘𝑏) → 𝑎 = 𝑏)))
95		dff13 7045	. . . . . . . . . 10 ⊢ (𝑧:(1...𝐾)–1-1→(1...𝑁) ↔ (𝑧:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑎 ∈ (1...𝐾)∀𝑏 ∈ (1...𝐾)((𝑧‘𝑎) = (𝑧‘𝑏) → 𝑎 = 𝑏)))
96	94, 95	sylibr 237	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧:(1...𝐾)–1-1→(1...𝑁))
97		hashf1rn 13884	. . . . . . . . 9 ⊢ (((1...𝐾) ∈ Fin ∧ 𝑧:(1...𝐾)–1-1→(1...𝑁)) → (♯‘𝑧) = (♯‘ran 𝑧))
98	32, 96, 97	syl2anc 587	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (♯‘𝑧) = (♯‘ran 𝑧))
99	31, 98	eqtrd 2771	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐾 = (♯‘ran 𝑧))
100	99	eqcomd 2742	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (♯‘ran 𝑧) = 𝐾)
101	1, 22, 100	elrabd 3593	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ran 𝑧 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾})
102		sticksstones2.3	. . . . . . 7 ⊢ 𝐵 = {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾}
103	102	eleq2i 2822	. . . . . 6 ⊢ (ran 𝑧 ∈ 𝐵 ↔ ran 𝑧 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾})
104	103	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (ran 𝑧 ∈ 𝐵 ↔ ran 𝑧 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾}))
105	101, 104	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ran 𝑧 ∈ 𝐵)
106		sticksstones2.5	. . . 4 ⊢ 𝐹 = (𝑧 ∈ 𝐴 ↦ ran 𝑧)
107	105, 106	fmptd 6909	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
108		sticksstones2.1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
109	108	3ad2ant1 1135	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) → 𝑁 ∈ ℕ₀)
110	109	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → 𝑁 ∈ ℕ₀)
111	26	3ad2ant1 1135	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) → 𝐾 ∈ ℕ₀)
112	111	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → 𝐾 ∈ ℕ₀)
113		simpl2 1194	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → 𝑖 ∈ 𝐴)
114		simpl3 1195	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → 𝑗 ∈ 𝐴)
115		simpr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → 𝑖 ≠ 𝑗)
116		fveq2 6695	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑠 → (𝑖‘𝑟) = (𝑖‘𝑠))
117		fveq2 6695	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑠 → (𝑗‘𝑟) = (𝑗‘𝑠))
118	116, 117	neeq12d 2993	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑠 → ((𝑖‘𝑟) ≠ (𝑗‘𝑟) ↔ (𝑖‘𝑠) ≠ (𝑗‘𝑠)))
119	118	cbvrabv 3392	. . . . . . . . . . 11 ⊢ {𝑟 ∈ (1...𝐾) ∣ (𝑖‘𝑟) ≠ (𝑗‘𝑟)} = {𝑠 ∈ (1...𝐾) ∣ (𝑖‘𝑠) ≠ (𝑗‘𝑠)}
120	119	infeq1i 9072	. . . . . . . . . 10 ⊢ inf({𝑟 ∈ (1...𝐾) ∣ (𝑖‘𝑟) ≠ (𝑗‘𝑟)}, ℝ, < ) = inf({𝑠 ∈ (1...𝐾) ∣ (𝑖‘𝑠) ≠ (𝑗‘𝑠)}, ℝ, < )
121	110, 112, 13, 113, 114, 115, 120	sticksstones1 39771	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → ran 𝑖 ≠ ran 𝑗)
122	106	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → 𝐹 = (𝑧 ∈ 𝐴 ↦ ran 𝑧))
123		simpr 488	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) ∧ 𝑧 = 𝑖) → 𝑧 = 𝑖)
124	123	rneqd 5792	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) ∧ 𝑧 = 𝑖) → ran 𝑧 = ran 𝑖)
125		fzfid 13511	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → (1...𝑁) ∈ Fin)
126		eleq1w 2813	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑖 → (𝑓 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴))
127		feq1 6504	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑖 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑖:(1...𝐾)⟶(1...𝑁)))
128		fveq1 6694	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 = 𝑖 → (𝑓‘𝑥) = (𝑖‘𝑥))
129		fveq1 6694	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 = 𝑖 → (𝑓‘𝑦) = (𝑖‘𝑦))
130	128, 129	breq12d 5052	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = 𝑖 → ((𝑓‘𝑥) < (𝑓‘𝑦) ↔ (𝑖‘𝑥) < (𝑖‘𝑦)))
131	130	imbi2d 344	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = 𝑖 → ((𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ (𝑥 < 𝑦 → (𝑖‘𝑥) < (𝑖‘𝑦))))
132	131	2ralbidv 3110	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑖 → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑖‘𝑥) < (𝑖‘𝑦))))
133	127, 132	anbi12d 634	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑖 → ((𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦))) ↔ (𝑖:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑖‘𝑥) < (𝑖‘𝑦)))))
134	126, 133	bibi12d 349	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = 𝑖 → ((𝑓 ∈ 𝐴 ↔ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))) ↔ (𝑖 ∈ 𝐴 ↔ (𝑖:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑖‘𝑥) < (𝑖‘𝑦))))))
135	134, 16	chvarvv 2008	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ 𝐴 ↔ (𝑖:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑖‘𝑥) < (𝑖‘𝑦))))
136	135	biimpi 219	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ 𝐴 → (𝑖:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑖‘𝑥) < (𝑖‘𝑦))))
137	136	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → (𝑖:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑖‘𝑥) < (𝑖‘𝑦))))
138	137	simpld 498	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝑖:(1...𝐾)⟶(1...𝑁))
139	138	3adant3 1134	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) → 𝑖:(1...𝐾)⟶(1...𝑁))
140	139	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → 𝑖:(1...𝐾)⟶(1...𝑁))
141	140	frnd 6531	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → ran 𝑖 ⊆ (1...𝑁))
142	125, 141	sselpwd 5204	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → ran 𝑖 ∈ 𝒫 (1...𝑁))
143	122, 124, 113, 142	fvmptd 6803	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → (𝐹‘𝑖) = ran 𝑖)
144		simpr 488	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) ∧ 𝑧 = 𝑗) → 𝑧 = 𝑗)
145	144	rneqd 5792	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) ∧ 𝑧 = 𝑗) → ran 𝑧 = ran 𝑗)
146		fzfid 13511	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
147	146	3ad2ant1 1135	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) → (1...𝑁) ∈ Fin)
148		eleq1w 2813	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑗 → (𝑓 ∈ 𝐴 ↔ 𝑗 ∈ 𝐴))
149		feq1 6504	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑗 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑗:(1...𝐾)⟶(1...𝑁)))
150		fveq1 6694	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 = 𝑗 → (𝑓‘𝑥) = (𝑗‘𝑥))
151		fveq1 6694	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 = 𝑗 → (𝑓‘𝑦) = (𝑗‘𝑦))
152	150, 151	breq12d 5052	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = 𝑗 → ((𝑓‘𝑥) < (𝑓‘𝑦) ↔ (𝑗‘𝑥) < (𝑗‘𝑦)))
153	152	imbi2d 344	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = 𝑗 → ((𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ (𝑥 < 𝑦 → (𝑗‘𝑥) < (𝑗‘𝑦))))
154	153	2ralbidv 3110	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑗 → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑗‘𝑥) < (𝑗‘𝑦))))
155	149, 154	anbi12d 634	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑗 → ((𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦))) ↔ (𝑗:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑗‘𝑥) < (𝑗‘𝑦)))))
156	148, 155	bibi12d 349	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = 𝑗 → ((𝑓 ∈ 𝐴 ↔ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))) ↔ (𝑗 ∈ 𝐴 ↔ (𝑗:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑗‘𝑥) < (𝑗‘𝑦))))))
157	156, 16	chvarvv 2008	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ 𝐴 ↔ (𝑗:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑗‘𝑥) < (𝑗‘𝑦))))
158	157	biimpi 219	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ 𝐴 → (𝑗:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑗‘𝑥) < (𝑗‘𝑦))))
159	158	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝑗:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑗‘𝑥) < (𝑗‘𝑦))))
160	159	simpld 498	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗:(1...𝐾)⟶(1...𝑁))
161	160	3adant2 1133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) → 𝑗:(1...𝐾)⟶(1...𝑁))
162	161	frnd 6531	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) → ran 𝑗 ⊆ (1...𝑁))
163	147, 162	sselpwd 5204	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) → ran 𝑗 ∈ 𝒫 (1...𝑁))
164	163	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → ran 𝑗 ∈ 𝒫 (1...𝑁))
165	122, 145, 114, 164	fvmptd 6803	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → (𝐹‘𝑗) = ran 𝑗)
166	121, 143, 165	3netr4d 3009	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → (𝐹‘𝑖) ≠ (𝐹‘𝑗))
167	166	ex 416	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) → (𝑖 ≠ 𝑗 → (𝐹‘𝑖) ≠ (𝐹‘𝑗)))
168	167	necon4d 2956	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) → ((𝐹‘𝑖) = (𝐹‘𝑗) → 𝑖 = 𝑗))
169	168	3expa 1120	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑗 ∈ 𝐴) → ((𝐹‘𝑖) = (𝐹‘𝑗) → 𝑖 = 𝑗))
170	169	ralrimiva 3095	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ∀𝑗 ∈ 𝐴 ((𝐹‘𝑖) = (𝐹‘𝑗) → 𝑖 = 𝑗))
171	170	ralrimiva 3095	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 ((𝐹‘𝑖) = (𝐹‘𝑗) → 𝑖 = 𝑗))
172	107, 171	jca 515	. 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ∧ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 ((𝐹‘𝑖) = (𝐹‘𝑗) → 𝑖 = 𝑗)))
173		dff13 7045	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 ((𝐹‘𝑖) = (𝐹‘𝑗) → 𝑖 = 𝑗)))
174	172, 173	sylibr 237	1 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 847 ∧ w3a 1089 ∀wal 1541 = wceq 1543 ∈ wcel 2112 {cab 2714 ≠ wne 2932 ∀wral 3051 {crab 3055 𝒫 cpw 4499 class class class wbr 5039 ↦ cmpt 5120 ran crn 5537 Fn wfn 6353 ⟶wf 6354 –1-1→wf1 6355 ‘cfv 6358 (class class class)co 7191 Fincfn 8604 infcinf 9035 ℝcr 10693 1c1 10695 < clt 10832 ℕcn 11795 ℕ₀cn0 12055 ...cfz 13060 ♯chash 13861

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-inf 9037 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-hash 13862

This theorem is referenced by: sticksstones3 39773 sticksstones4 39774

W3C validator