sticksstones2 - Mathbox for metakunt

	Mathbox for metakunt		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > sticksstones2		Structured version Visualization version GIF version

Theorem sticksstones2 42974

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 6894	. . . . . 6 ⊢ (𝑎 = ran 𝑧 → ((♯‘𝑎) = 𝐾 ↔ (♯‘ran 𝑧) = 𝐾))
2		fzfid 14029	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (1...𝑁) ∈ Fin)
3		eleq1w 2848	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑧 → (𝑓 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
4		feq1 6687	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑧 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑧:(1...𝐾)⟶(1...𝑁)))
5		fveq1 6884	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝑧 → (𝑓‘𝑥) = (𝑧‘𝑥))
6		fveq1 6884	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝑧 → (𝑓‘𝑦) = (𝑧‘𝑦))
7	5, 6	breq12d 5124	. . . . . . . . . . . . . . . 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 3190	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑧 → (∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ ∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑧‘𝑥) < (𝑧‘𝑦))))
10	9	ralbidv 3190	. . . . . . . . . . . . 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 2904	. . . . . . . . . . . . 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 2223	. . . . . . . . . . 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 6718	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ran 𝑧 ⊆ (1...𝑁))
21	2, 20	sselpwd 5301	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ran 𝑧 ∈ 𝒫 (1...𝑁))
22	19	ffnd 6710	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 Fn (1...𝐾))
23		hashfn 14431	. . . . . . . . . . 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 14402	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℕ₀ → (♯‘(1...𝐾)) = 𝐾)
28	26, 27	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (♯‘(1...𝐾)) = 𝐾)
29	24, 28	eqtrd 2800	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (♯‘𝑧) = 𝐾)
30	29	eqcomd 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐾 = (♯‘𝑧))
31		fzfid 14029	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (1...𝐾) ∈ Fin)
32		elfznn 13600	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ (1...𝐾) → 𝑎 ∈ ℕ)
33	32	3ad2ant3 1153	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) → 𝑎 ∈ ℕ)
34	33	nnred 12265	. . . . . . . . . . . . . . . . . 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 13600	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∈ (1...𝐾) → 𝑏 ∈ ℕ)
37	36	nnred 12265	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ (1...𝐾) → 𝑏 ∈ ℝ)
38	37	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → 𝑏 ∈ ℝ)
39		lttri2 11309	. . . . . . . . . . . . . . . . 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 7084	. . . . . . . . . . . . . . . . . . . . . . 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 13600	. . . . . . . . . . . . . . . . . . . . 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 12265	. . . . . . . . . . . . . . . . . . 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 5114	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑎 → (𝑥 < 𝑦 ↔ 𝑎 < 𝑦))
55		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = 𝑎 → (𝑧‘𝑥) = (𝑧‘𝑎))
56	55	breq1d 5121	. . . . . . . . . . . . . . . . . . . . . . . 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 5115	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑏 → (𝑎 < 𝑦 ↔ 𝑎 < 𝑏))
59		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = 𝑏 → (𝑧‘𝑦) = (𝑧‘𝑏))
60	59	breq2d 5123	. . . . . . . . . . . . . . . . . . . . . . . 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 3594	. . . . . . . . . . . . . . . . . . . . . 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 11363	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑎 < 𝑏) → (𝑧‘𝑎) ≠ (𝑧‘𝑏))
67	41	ffvelcdmda 7083	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → (𝑧‘𝑏) ∈ (1...𝑁))
68		elfznn 13600	. . . . . . . . . . . . . . . . . . . . . . 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 12265	. . . . . . . . . . . . . . . . . . . . 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 5114	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = 𝑏 → (𝑥 < 𝑦 ↔ 𝑏 < 𝑦))
73		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = 𝑏 → (𝑧‘𝑥) = (𝑧‘𝑏))
74	73	breq1d 5121	. . . . . . . . . . . . . . . . . . . . . . . . 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 5115	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = 𝑎 → (𝑏 < 𝑦 ↔ 𝑏 < 𝑎))
77		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 = 𝑎 → (𝑧‘𝑦) = (𝑧‘𝑎))
78	77	breq2d 5123	. . . . . . . . . . . . . . . . . . . . . . . . 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 3594	. . . . . . . . . . . . . . . . . . . . . . 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 11363	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑏 < 𝑎) → (𝑧‘𝑏) ≠ (𝑧‘𝑎))
85	84	necomd 3015	. . . . . . . . . . . . . . . . . 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 2984	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) → ((𝑧‘𝑎) = (𝑧‘𝑏) → 𝑎 = 𝑏))
90	89	ralrimiva 3159	. . . . . . . . . . . . 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 3159	. . . . . . . . . . 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 7257	. . . . . . . . . 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 14408	. . . . . . . . 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 2800	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐾 = (♯‘ran 𝑧))
99	98	eqcomd 2771	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (♯‘ran 𝑧) = 𝐾)
100	1, 21, 99	elrabd 3654	. . . . 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 2857	. . . . . 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 7113	. . 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 6885	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑠 → (𝑖‘𝑟) = (𝑖‘𝑠))
116		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑠 → (𝑗‘𝑟) = (𝑗‘𝑠))
117	115, 116	neeq12d 3021	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑠 → ((𝑖‘𝑟) ≠ (𝑗‘𝑟) ↔ (𝑖‘𝑠) ≠ (𝑗‘𝑠)))
118	117	cbvrabv 3428	. . . . . . . . . . 11 ⊢ {𝑟 ∈ (1...𝐾) ∣ (𝑖‘𝑟) ≠ (𝑗‘𝑟)} = {𝑠 ∈ (1...𝐾) ∣ (𝑖‘𝑠) ≠ (𝑗‘𝑠)}
119	118	infeq1i 9446	. . . . . . . . . 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 42973	. . . . . . . . 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 5930	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) ∧ 𝑧 = 𝑖) → ran 𝑧 = ran 𝑖)
124		fzfid 14029	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → (1...𝑁) ∈ Fin)
125		eleq1w 2848	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = 𝑖 → (𝑓 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴))
126		feq1 6687	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑖 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑖:(1...𝐾)⟶(1...𝑁)))
127		fveq1 6884	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = 𝑖 → (𝑓‘𝑥) = (𝑖‘𝑥))
128		fveq1 6884	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = 𝑖 → (𝑓‘𝑦) = (𝑖‘𝑦))
129	127, 128	breq12d 5124	. . . . . . . . . . . . . . . . . . . . 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 3231	. . . . . . . . . . . . . . . . . . 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 6718	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → ran 𝑖 ⊆ (1...𝑁))
140	124, 139	sselpwd 5301	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → ran 𝑖 ∈ 𝒫 (1...𝑁))
141	121, 123, 112, 140	fvmptd 7001	. . . . . . . . 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 5930	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) ∧ 𝑧 = 𝑗) → ran 𝑧 = ran 𝑗)
144		fzfid 14029	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
145	144	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) → (1...𝑁) ∈ Fin)
146		eleq1w 2848	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = 𝑗 → (𝑓 ∈ 𝐴 ↔ 𝑗 ∈ 𝐴))
147		feq1 6687	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑗 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑗:(1...𝐾)⟶(1...𝑁)))
148		fveq1 6884	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = 𝑗 → (𝑓‘𝑥) = (𝑗‘𝑥))
149		fveq1 6884	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = 𝑗 → (𝑓‘𝑦) = (𝑗‘𝑦))
150	148, 149	breq12d 5124	. . . . . . . . . . . . . . . . . . . . 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 3231	. . . . . . . . . . . . . . . . . . 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 6718	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) → ran 𝑗 ⊆ (1...𝑁))
160	145, 159	sselpwd 5301	. . . . . . . . . . 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 7001	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 ≠ 𝑗) → (𝐹‘𝑗) = ran 𝑗)
163	120, 141, 162	3netr4d 3037	. . . . . . . 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 2984	. . . . . 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 3159	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ∀𝑗 ∈ 𝐴 ((𝐹‘𝑖) = (𝐹‘𝑗) → 𝑖 = 𝑗))
168	167	ralrimiva 3159	. . 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 7257	. 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 2146 {cab 2743 ≠ wne 2960 ∀wral 3081 {crab 3418 𝒫 cpw 4564 class class class wbr 5111 ↦ cmpt 5194 ran crn 5664 Fn wfn 6535 ⟶wf 6536 –1-1→wf1 6537 ‘cfv 6540 (class class class)co 7419 Fincfn 8949 infcinf 9408 ℝcr 11116 1c1 11118 < clt 11260 ℕcn 12250 ℕ₀cn0 12521 ...cfz 13553 ♯chash 14386

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194 ax-pre-sup 11195

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-sup 9409 df-inf 9410 df-card 9941 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-nn 12251 df-n0 12522 df-z 12609 df-uz 12881 df-fz 13554 df-hash 14387

This theorem is used by: sticksstones3 42975 sticksstones4 42976

W3C validator