sticksstones2 - Mathbox for metakunt

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Theorem sticksstones2 43014

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 6888	. . . . . 6 ⊢ (𝑎 = ran 𝑧 → ((♯‘𝑎) = 𝐾 ↔ (♯‘ran 𝑧) = 𝐾))
2		fzfid 14038	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (1...𝑁) ∈ Fin)
3		eleq1w 2843	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑧 → (𝑓 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
4		feq1 6681	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑧 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑧:(1...𝐾)⟶(1...𝑁)))
5		fveq1 6878	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝑧 → (𝑓‘𝑥) = (𝑧‘𝑥))
6		fveq1 6878	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝑧 → (𝑓‘𝑦) = (𝑧‘𝑦))
7	5, 6	breq12d 5116	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑧 → ((𝑓‘𝑥) < (𝑓‘𝑦) ↔ (𝑧‘𝑥) < (𝑧‘𝑦)))
8	7	imbi2d 343	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑧 → ((𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ (𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦))))
9	8	ralbidv 3185	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑧 → (∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ ∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦))))
10	9	ralbidv 3185	. . . . . . . . . . . . 13 ⊢ (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 644	. . . . . . . . . . . 12 ⊢ (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 348	. . . . . . . . . . 11 ⊢ (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	. . . . . . . . . . . . 13 ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}
14		eqabb 2899	. . . . . . . . . . . . 13 ⊢ (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	. . . . . . . . . . . 12 ⊢ ∀𝑓(𝑓 ∈ 𝐴 ↔ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦))))
16	15	spi 2220	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝐴 ↔ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦))))
17	12, 16	chvarvv 2022	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐴 ↔ (𝑧:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦))))
18	17	bilani 510	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝑧:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦))))
19	18	simpld 500	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧:(1...𝐾)⟶(1...𝑁))
20	19	frnd 6712	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ran 𝑧 ⊆ (1...𝑁))
21	2, 20	sselpwd 5293	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ran 𝑧 ∈ 𝒫 (1...𝑁))
22	19	ffnd 6704	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 Fn (1...𝐾))
23		hashfn 14440	. . . . . . . . . . 11 ⊢ (𝑧 Fn (1...𝐾) → (♯‘𝑧) = (♯‘(1...𝐾)))
24	22, 23	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (♯‘𝑧) = (♯‘(1...𝐾)))
25		sticksstones2.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
26	25	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐾 ∈ ℕ₀)
27		hashfz1 14411	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℕ₀ → (♯‘(1...𝐾)) = 𝐾)
28	26, 27	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (♯‘(1...𝐾)) = 𝐾)
29	24, 28	eqtrd 2795	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (♯‘𝑧) = 𝐾)
30	29	eqcomd 2766	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐾 = (♯‘𝑧))
31		fzfid 14038	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (1...𝐾) ∈ Fin)
32		elfznn 13609	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ (1...𝐾) → 𝑎 ∈ ℕ)
33	32	3ad2ant3 1153	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) → 𝑎 ∈ ℕ)
34	33	nnred 12273	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) → 𝑎 ∈ ℝ)
35	34	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → 𝑎 ∈ ℝ)
36		elfznn 13609	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∈ (1...𝐾) → 𝑏 ∈ ℕ)
37	36	nnred 12273	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ (1...𝐾) → 𝑏 ∈ ℝ)
38	37	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → 𝑏 ∈ ℝ)
39		lttri2 11317	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎 ≠ 𝑏 ↔ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎)))
40	35, 38, 39	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑎 ≠ 𝑏 ↔ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎)))
41	19	3adant3 1150	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) → 𝑧:(1...𝐾)⟶(1...𝑁))
42		simp3 1156	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) → 𝑎 ∈ (1...𝐾))
43	41, 42	ffvelcdmd 7079	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) → (𝑧‘𝑎) ∈ (1...𝑁))
44	43	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑧‘𝑎) ∈ (1...𝑁))
45	44	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑎 < 𝑏) → (𝑧‘𝑎) ∈ (1...𝑁))
46		elfznn 13609	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧‘𝑎) ∈ (1...𝑁) → (𝑧‘𝑎) ∈ ℕ)
47	45, 46	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑎 < 𝑏) → (𝑧‘𝑎) ∈ ℕ)
48	47	nnred 12273	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑎 < 𝑏) → (𝑧‘𝑎) ∈ ℝ)
49	18	simprd 501	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦)))
50	49	3adant3 1150	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦)))
51	50	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦)))
52	42	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → 𝑎 ∈ (1...𝐾))
53		simpr 490	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → 𝑏 ∈ (1...𝐾))
54		breq1 5106	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑎 → (𝑥 < 𝑦 ↔ 𝑎 < 𝑦))
55		fveq2 6879	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = 𝑎 → (𝑧‘𝑥) = (𝑧‘𝑎))
56	55	breq1d 5113	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑎 → ((𝑧‘𝑥) < (𝑧‘𝑦) ↔ (𝑧‘𝑎) < (𝑧‘𝑦)))
57	54, 56	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑎 → ((𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦)) ↔ (𝑎 < 𝑦 → (𝑧‘𝑎) < (𝑧‘𝑦))))
58		breq2 5107	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑏 → (𝑎 < 𝑦 ↔ 𝑎 < 𝑏))
59		fveq2 6879	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = 𝑏 → (𝑧‘𝑦) = (𝑧‘𝑏))
60	59	breq2d 5115	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑏 → ((𝑧‘𝑎) < (𝑧‘𝑦) ↔ (𝑧‘𝑎) < (𝑧‘𝑏)))
61	58, 60	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑏 → ((𝑎 < 𝑦 → (𝑧‘𝑎) < (𝑧‘𝑦)) ↔ (𝑎 < 𝑏 → (𝑧‘𝑎) < (𝑧‘𝑏))))
62	57, 61	rspc2v 3587	. . . . . . . . . . . . . . . . . . . . . 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))))
63	52, 53, 62	syl2anc 596	. . . . . . . . . . . . . . . . . . . . 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))))
64	51, 63	mpd 16	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑎 < 𝑏 → (𝑧‘𝑎) < (𝑧‘𝑏)))
65	64	imp 412	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑎 < 𝑏) → (𝑧‘𝑎) < (𝑧‘𝑏))
66	48, 65	ltned 11371	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑎 < 𝑏) → (𝑧‘𝑎) ≠ (𝑧‘𝑏))
67	41	ffvelcdmda 7078	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑧‘𝑏) ∈ (1...𝑁))
68		elfznn 13609	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧‘𝑏) ∈ (1...𝑁) → (𝑧‘𝑏) ∈ ℕ)
69	67, 68	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑧‘𝑏) ∈ ℕ)
70	69	nnred 12273	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑧‘𝑏) ∈ ℝ)
71	70	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑏 < 𝑎) → (𝑧‘𝑏) ∈ ℝ)
72		breq1 5106	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = 𝑏 → (𝑥 < 𝑦 ↔ 𝑏 < 𝑦))
73		fveq2 6879	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = 𝑏 → (𝑧‘𝑥) = (𝑧‘𝑏))
74	73	breq1d 5113	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = 𝑏 → ((𝑧‘𝑥) < (𝑧‘𝑦) ↔ (𝑧‘𝑏) < (𝑧‘𝑦)))
75	72, 74	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑏 → ((𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦)) ↔ (𝑏 < 𝑦 → (𝑧‘𝑏) < (𝑧‘𝑦))))
76		breq2 5107	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = 𝑎 → (𝑏 < 𝑦 ↔ 𝑏 < 𝑎))
77		fveq2 6879	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 = 𝑎 → (𝑧‘𝑦) = (𝑧‘𝑎))
78	77	breq2d 5115	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = 𝑎 → ((𝑧‘𝑏) < (𝑧‘𝑦) ↔ (𝑧‘𝑏) < (𝑧‘𝑎)))
79	76, 78	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑎 → ((𝑏 < 𝑦 → (𝑧‘𝑏) < (𝑧‘𝑦)) ↔ (𝑏 < 𝑎 → (𝑧‘𝑏) < (𝑧‘𝑎))))
80	75, 79	rspc2v 3587	. . . . . . . . . . . . . . . . . . . . . . 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))))
81	53, 52, 80	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . 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))))
82	51, 81	mpd 16	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑏 < 𝑎 → (𝑧‘𝑏) < (𝑧‘𝑎)))
83	82	imp 412	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑏 < 𝑎) → (𝑧‘𝑏) < (𝑧‘𝑎))
84	71, 83	ltned 11371	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑏 < 𝑎) → (𝑧‘𝑏) ≠ (𝑧‘𝑎))
85	84	necomd 3010	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑏 < 𝑎) → (𝑧‘𝑎) ≠ (𝑧‘𝑏))
86	66, 85	jaodan 972	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎)) → (𝑧‘𝑎) ≠ (𝑧‘𝑏))
87	86	ex 418	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → ((𝑎 < 𝑏 ∨ 𝑏 < 𝑎) → (𝑧‘𝑎) ≠ (𝑧‘𝑏)))
88	40, 87	sylbid 243	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑎 ≠ 𝑏 → (𝑧‘𝑎) ≠ (𝑧‘𝑏)))
89	88	necon4d 2979	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → ((𝑧‘𝑎) = (𝑧‘𝑏) → 𝑎 = 𝑏))
90	89	ralrimiva 3154	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) → ∀𝑏 ∈ (1...𝐾)((𝑧‘𝑎) = (𝑧‘𝑏) → 𝑎 = 𝑏))
91	90	3expa 1136	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑎 ∈ (1...𝐾)) → ∀𝑏 ∈ (1...𝐾)((𝑧‘𝑎) = (𝑧‘𝑏) → 𝑎 = 𝑏))
92	91	ralrimiva 3154	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ∀𝑎 ∈ (1...𝐾)∀𝑏 ∈ (1...𝐾)((𝑧‘𝑎) = (𝑧‘𝑏) → 𝑎 = 𝑏))
93	19, 92	jca 521	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝑧:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑎 ∈ (1...𝐾)∀𝑏 ∈ (1...𝐾)((𝑧‘𝑎) = (𝑧‘𝑏) → 𝑎 = 𝑏)))
94		dff13 7252	. . . . . . . . . 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)))
95	93, 94	sylibr 237	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧:(1...𝐾)–1-1→(1...𝑁))
96		hashf1rn 14417	. . . . . . . . 9 ⊢ (((1...𝐾) ∈ Fin ∧ 𝑧:(1...𝐾)–1-1→(1...𝑁)) → (♯‘𝑧) = (♯‘ran 𝑧))
97	31, 95, 96	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (♯‘𝑧) = (♯‘ran 𝑧))
98	30, 97	eqtrd 2795	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐾 = (♯‘ran 𝑧))
99	98	eqcomd 2766	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (♯‘ran 𝑧) = 𝐾)
100	1, 21, 99	elrabd 3647	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ran 𝑧 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾})
101		sticksstones2.3	. . . . . . 7 ⊢ 𝐵 = {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾}
102	101	eleq2i 2852	. . . . . 6 ⊢ (ran 𝑧 ∈ 𝐵 ↔ ran 𝑧 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾})
103	102	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (ran 𝑧 ∈ 𝐵 ↔ ran 𝑧 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾}))
104	100, 103	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ran 𝑧 ∈ 𝐵)
105		sticksstones2.5	. . . 4 ⊢ 𝐹 = (𝑧 ∈ 𝐴 ↦ ran 𝑧)
106	104, 105	fmptd 7108	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
107		sticksstones2.1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
108	107	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) → 𝑁 ∈ ℕ₀)
109	108	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → 𝑁 ∈ ℕ₀)
110	25	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) → 𝐾 ∈ ℕ₀)
111	110	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → 𝐾 ∈ ℕ₀)
112		simpl2 1211	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → 𝑖 ∈ 𝐴)
113		simpl3 1212	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → 𝑗 ∈ 𝐴)
114		simpr 490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → 𝑖 ≠ 𝑗)
115		fveq2 6879	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑠 → (𝑖‘𝑟) = (𝑖‘𝑠))
116		fveq2 6879	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑠 → (𝑗‘𝑟) = (𝑗‘𝑠))
117	115, 116	neeq12d 3016	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑠 → ((𝑖‘𝑟) ≠ (𝑗‘𝑟) ↔ (𝑖‘𝑠) ≠ (𝑗‘𝑠)))
118	117	cbvrabv 3422	. . . . . . . . . . 11 ⊢ {𝑟 ∈ (1...𝐾) ∣ (𝑖‘𝑟) ≠ (𝑗‘𝑟)} = {𝑠 ∈ (1...𝐾) ∣ (𝑖‘𝑠) ≠ (𝑗‘𝑠)}
119	118	infeq1i 9450	. . . . . . . . . 10 ⊢ inf({𝑟 ∈ (1...𝐾) ∣ (𝑖‘𝑟) ≠ (𝑗‘𝑟)}, ℝ, < ) = inf({𝑠 ∈ (1...𝐾) ∣ (𝑖‘𝑠) ≠ (𝑗‘𝑠)}, ℝ, < )
120	109, 111, 13, 112, 113, 114, 119	sticksstones1 43013	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → ran 𝑖 ≠ ran 𝑗)
121	105	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → 𝐹 = (𝑧 ∈ 𝐴 ↦ ran 𝑧))
122		simpr 490	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) ∧ 𝑧 = 𝑖) → 𝑧 = 𝑖)
123	122	rneqd 5922	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) ∧ 𝑧 = 𝑖) → ran 𝑧 = ran 𝑖)
124		fzfid 14038	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → (1...𝑁) ∈ Fin)
125		eleq1w 2843	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = 𝑖 → (𝑓 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴))
126		feq1 6681	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑖 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑖:(1...𝐾)⟶(1...𝑁)))
127		fveq1 6878	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = 𝑖 → (𝑓‘𝑥) = (𝑖‘𝑥))
128		fveq1 6878	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = 𝑖 → (𝑓‘𝑦) = (𝑖‘𝑦))
129	127, 128	breq12d 5116	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = 𝑖 → ((𝑓‘𝑥) < (𝑓‘𝑦) ↔ (𝑖‘𝑥) < (𝑖‘𝑦)))
130	129	imbi2d 343	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑖 → ((𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ (𝑥 < 𝑦 → (𝑖‘𝑥) < (𝑖‘𝑦))))
131	130	2ralbidv 3226	. . . . . . . . . . . . . . . . . . 19 ⊢ (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))))
132	126, 131	anbi12d 644	. . . . . . . . . . . . . . . . . 18 ⊢ (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)))))
133	125, 132	bibi12d 348	. . . . . . . . . . . . . . . . 17 ⊢ (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))))))
134	133, 16	chvarvv 2022	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ 𝐴 ↔ (𝑖:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑖‘𝑥) < (𝑖‘𝑦))))
135	134	bilani 510	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → (𝑖:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑖‘𝑥) < (𝑖‘𝑦))))
136	135	simpld 500	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝑖:(1...𝐾)⟶(1...𝑁))
137	136	3adant3 1150	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) → 𝑖:(1...𝐾)⟶(1...𝑁))
138	137	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → 𝑖:(1...𝐾)⟶(1...𝑁))
139	138	frnd 6712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → ran 𝑖 ⊆ (1...𝑁))
140	124, 139	sselpwd 5293	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → ran 𝑖 ∈ 𝒫 (1...𝑁))
141	121, 123, 112, 140	fvmptd 6995	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → (𝐹‘𝑖) = ran 𝑖)
142		simpr 490	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) ∧ 𝑧 = 𝑗) → 𝑧 = 𝑗)
143	142	rneqd 5922	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) ∧ 𝑧 = 𝑗) → ran 𝑧 = ran 𝑗)
144		fzfid 14038	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
145	144	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) → (1...𝑁) ∈ Fin)
146		eleq1w 2843	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = 𝑗 → (𝑓 ∈ 𝐴 ↔ 𝑗 ∈ 𝐴))
147		feq1 6681	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑗 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑗:(1...𝐾)⟶(1...𝑁)))
148		fveq1 6878	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = 𝑗 → (𝑓‘𝑥) = (𝑗‘𝑥))
149		fveq1 6878	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = 𝑗 → (𝑓‘𝑦) = (𝑗‘𝑦))
150	148, 149	breq12d 5116	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = 𝑗 → ((𝑓‘𝑥) < (𝑓‘𝑦) ↔ (𝑗‘𝑥) < (𝑗‘𝑦)))
151	150	imbi2d 343	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑗 → ((𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ (𝑥 < 𝑦 → (𝑗‘𝑥) < (𝑗‘𝑦))))
152	151	2ralbidv 3226	. . . . . . . . . . . . . . . . . . 19 ⊢ (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))))
153	147, 152	anbi12d 644	. . . . . . . . . . . . . . . . . 18 ⊢ (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)))))
154	146, 153	bibi12d 348	. . . . . . . . . . . . . . . . 17 ⊢ (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))))))
155	154, 16	chvarvv 2022	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ 𝐴 ↔ (𝑗:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑗‘𝑥) < (𝑗‘𝑦))))
156	155	bilani 510	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝑗:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑗‘𝑥) < (𝑗‘𝑦))))
157	156	simpld 500	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗:(1...𝐾)⟶(1...𝑁))
158	157	3adant2 1149	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) → 𝑗:(1...𝐾)⟶(1...𝑁))
159	158	frnd 6712	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) → ran 𝑗 ⊆ (1...𝑁))
160	145, 159	sselpwd 5293	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) → ran 𝑗 ∈ 𝒫 (1...𝑁))
161	160	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → ran 𝑗 ∈ 𝒫 (1...𝑁))
162	121, 143, 113, 161	fvmptd 6995	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → (𝐹‘𝑗) = ran 𝑗)
163	120, 141, 162	3netr4d 3032	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → (𝐹‘𝑖) ≠ (𝐹‘𝑗))
164	163	ex 418	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) → (𝑖 ≠ 𝑗 → (𝐹‘𝑖) ≠ (𝐹‘𝑗)))
165	164	necon4d 2979	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) → ((𝐹‘𝑖) = (𝐹‘𝑗) → 𝑖 = 𝑗))
166	165	3expa 1136	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑗 ∈ 𝐴) → ((𝐹‘𝑖) = (𝐹‘𝑗) → 𝑖 = 𝑗))
167	166	ralrimiva 3154	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ∀𝑗 ∈ 𝐴 ((𝐹‘𝑖) = (𝐹‘𝑗) → 𝑖 = 𝑗))
168	167	ralrimiva 3154	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 ((𝐹‘𝑖) = (𝐹‘𝑗) → 𝑖 = 𝑗))
169	106, 168	jca 521	. 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ∧ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 ((𝐹‘𝑖) = (𝐹‘𝑗) → 𝑖 = 𝑗)))
170		dff13 7252	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 ((𝐹‘𝑖) = (𝐹‘𝑗) → 𝑖 = 𝑗)))
171	169, 170	sylibr 237	1 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 ∀wal 1568 = wceq 1570 ∈ wcel 2145 {cab 2738 ≠ wne 2955 ∀wral 3076 {crab 3412 𝒫 cpw 4557 class class class wbr 5103 ↦ cmpt 5186 ran crn 5656 Fn wfn 6528 ⟶wf 6529 –1-1→wf1 6530 ‘cfv 6533 (class class class)co 7414 Fincfn 8953 infcinf 9412 ℝcr 11124 1c1 11126 < clt 11268 ℕcn 12258 ℕ₀cn0 12529 ...cfz 13562 ♯chash 14395

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-er 8697 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9413 df-inf 9414 df-card 9945 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-nn 12259 df-n0 12530 df-z 12617 df-uz 12889 df-fz 13563 df-hash 14396

This theorem is used by: sticksstones3 43015 sticksstones4 43016

W3C validator