sticksstones3 - Mathbox for metakunt

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Theorem sticksstones3 43017

Description: The range function on strictly monotone functions with finite domain and codomain is an surjective mapping onto 𝐾-elemental sets. (Contributed by metakunt, 28-Sep-2024.)

**Hypotheses**
Ref	Expression
sticksstones3.1	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
sticksstones3.2	⊢ (𝜑 → 𝐾 ∈ ℕ₀)
sticksstones3.3	⊢ 𝐵 = {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾}
sticksstones3.4	⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}
sticksstones3.5	⊢ 𝐹 = (𝑧 ∈ 𝐴 ↦ ran 𝑧)

**Assertion**
Ref	Expression
sticksstones3	⊢ (𝜑 → 𝐹:𝐴–onto→𝐵)

Distinct variable groups: 𝐴,𝑎 𝐴,𝑓,𝑧 𝑥,𝐵,𝑦,𝑧 𝐾,𝑎,𝑥,𝑦 𝑓,𝐾,𝑥,𝑦 𝑁,𝑎 𝑓,𝑁 𝜑,𝑎,𝑥,𝑦,𝑧 𝜑,𝑓 Allowed substitution hints: 𝐴(𝑥, 𝑦) 𝐵(𝑓, 𝑎) 𝐹(𝑥, 𝑦, 𝑧, 𝑓, 𝑎) 𝐾(𝑧) 𝑁(𝑥, 𝑦, 𝑧)

Proof of Theorem sticksstones3 Dummy variables 𝑤 𝑣 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sticksstones3.1	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
2		sticksstones3.2	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
3		sticksstones3.3	. . . . 5 ⊢ 𝐵 = {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾}
4		sticksstones3.4	. . . . 5 ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}
5		sticksstones3.5	. . . . 5 ⊢ 𝐹 = (𝑧 ∈ 𝐴 ↦ ran 𝑧)
6	1, 2, 3, 4, 5	sticksstones2 43016	. . . 4 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)
7		df-f1 6538	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ^◡𝐹))
8	7	biimpi 219	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹:𝐴⟶𝐵 ∧ Fun ^◡𝐹))
9	8	simpld 500	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
10	6, 9	syl 18	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
11	3	eleq2i 2852	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ 𝐵 ↔ 𝑤 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾})
12	11	bilani 510	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾})
13		fveqeq2 6888	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑤 → ((♯‘𝑎) = 𝐾 ↔ (♯‘𝑤) = 𝐾))
14	13	elrab 3645	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾} ↔ (𝑤 ∈ 𝒫 (1...𝑁) ∧ (♯‘𝑤) = 𝐾))
15	12, 14	sylib 221	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝑤 ∈ 𝒫 (1...𝑁) ∧ (♯‘𝑤) = 𝐾))
16	15	simpld 500	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ 𝒫 (1...𝑁))
17	16	elpwid 4566	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ⊆ (1...𝑁))
18	17	sseld 3930	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝑐 ∈ 𝑤 → 𝑐 ∈ (1...𝑁)))
19	18	imp 412	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ 𝑐 ∈ 𝑤) → 𝑐 ∈ (1...𝑁))
20	19	3impa 1127	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑐 ∈ 𝑤) → 𝑐 ∈ (1...𝑁))
21		elfznn 13611	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (1...𝑁) → 𝑐 ∈ ℕ)
22	20, 21	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑐 ∈ 𝑤) → 𝑐 ∈ ℕ)
23	22	nnred 12275	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑐 ∈ 𝑤) → 𝑐 ∈ ℝ)
24	23	3expa 1136	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ 𝑐 ∈ 𝑤) → 𝑐 ∈ ℝ)
25	24	ex 418	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝑐 ∈ 𝑤 → 𝑐 ∈ ℝ))
26	25	ssrdv 3937	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ⊆ ℝ)
27		ltso 11317	. . . . . . . . 9 ⊢ < Or ℝ
28		soss 5583	. . . . . . . . 9 ⊢ (𝑤 ⊆ ℝ → ( < Or ℝ → < Or 𝑤))
29	27, 28	mpi 21	. . . . . . . 8 ⊢ (𝑤 ⊆ ℝ → < Or 𝑤)
30	26, 29	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → < Or 𝑤)
31		fzfid 14040	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (1...𝑁) ∈ Fin)
32	31, 17	ssfid 9242	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ Fin)
33		fz1iso 14530	. . . . . . 7 ⊢ (( < Or 𝑤 ∧ 𝑤 ∈ Fin) → ∃𝑣 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤))
34	30, 32, 33	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → ∃𝑣 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤))
35		df-isom 6542	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤) ↔ (𝑣:(1...(♯‘𝑤))–1-1-onto→𝑤 ∧ ∀𝑥 ∈ (1...(♯‘𝑤))∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 ↔ (𝑣‘𝑥) < (𝑣‘𝑦))))
36	35	biimpi 219	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤) → (𝑣:(1...(♯‘𝑤))–1-1-onto→𝑤 ∧ ∀𝑥 ∈ (1...(♯‘𝑤))∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 ↔ (𝑣‘𝑥) < (𝑣‘𝑦))))
37	36	3ad2ant3 1153	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣:(1...(♯‘𝑤))–1-1-onto→𝑤 ∧ ∀𝑥 ∈ (1...(♯‘𝑤))∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 ↔ (𝑣‘𝑥) < (𝑣‘𝑦))))
38	37	simpld 500	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣:(1...(♯‘𝑤))–1-1-onto→𝑤)
39	15	simprd 501	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (♯‘𝑤) = 𝐾)
40		oveq2 7422	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑤) = 𝐾 → (1...(♯‘𝑤)) = (1...𝐾))
41	40	f1oeq2d 6814	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝑤) = 𝐾 → (𝑣:(1...(♯‘𝑤))–1-1-onto→𝑤 ↔ 𝑣:(1...𝐾)–1-1-onto→𝑤))
42	39, 41	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝑣:(1...(♯‘𝑤))–1-1-onto→𝑤 ↔ 𝑣:(1...𝐾)–1-1-onto→𝑤))
43	42	biimpd 232	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝑣:(1...(♯‘𝑤))–1-1-onto→𝑤 → 𝑣:(1...𝐾)–1-1-onto→𝑤))
44	43	3adant3 1150	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣:(1...(♯‘𝑤))–1-1-onto→𝑤 → 𝑣:(1...𝐾)–1-1-onto→𝑤))
45	38, 44	mpd 16	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣:(1...𝐾)–1-1-onto→𝑤)
46		f1of 6818	. . . . . . . . . . . . . . 15 ⊢ (𝑣:(1...𝐾)–1-1-onto→𝑤 → 𝑣:(1...𝐾)⟶𝑤)
47	45, 46	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣:(1...𝐾)⟶𝑤)
48	47	ffnd 6704	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣 Fn (1...𝐾))
49		ovexd 7449	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (1...𝐾) ∈ V)
50	48, 49	fnexd 7218	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣 ∈ V)
51	17	3adant3 1150	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑤 ⊆ (1...𝑁))
52		fss 6720	. . . . . . . . . . . . . 14 ⊢ ((𝑣:(1...𝐾)⟶𝑤 ∧ 𝑤 ⊆ (1...𝑁)) → 𝑣:(1...𝐾)⟶(1...𝑁))
53	47, 51, 52	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣:(1...𝐾)⟶(1...𝑁))
54	37	simprd 501	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → ∀𝑥 ∈ (1...(♯‘𝑤))∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 ↔ (𝑣‘𝑥) < (𝑣‘𝑦)))
55		biimp 218	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 < 𝑦 ↔ (𝑣‘𝑥) < (𝑣‘𝑦)) → (𝑥 < 𝑦 → (𝑣‘𝑥) < (𝑣‘𝑦)))
56	55	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) ∧ 𝑥 ∈ (1...(♯‘𝑤))) ∧ 𝑦 ∈ (1...(♯‘𝑤))) → ((𝑥 < 𝑦 ↔ (𝑣‘𝑥) < (𝑣‘𝑦)) → (𝑥 < 𝑦 → (𝑣‘𝑥) < (𝑣‘𝑦))))
57	56	ralimdva 3174	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) ∧ 𝑥 ∈ (1...(♯‘𝑤))) → (∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 ↔ (𝑣‘𝑥) < (𝑣‘𝑦)) → ∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 → (𝑣‘𝑥) < (𝑣‘𝑦))))
58	57	ralimdva 3174	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (∀𝑥 ∈ (1...(♯‘𝑤))∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 ↔ (𝑣‘𝑥) < (𝑣‘𝑦)) → ∀𝑥 ∈ (1...(♯‘𝑤))∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 → (𝑣‘𝑥) < (𝑣‘𝑦))))
59	54, 58	mpd 16	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → ∀𝑥 ∈ (1...(♯‘𝑤))∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 → (𝑣‘𝑥) < (𝑣‘𝑦)))
60	39	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (♯‘𝑤) = 𝐾)
61	60	3impa 1127	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (♯‘𝑤) = 𝐾)
62	61	oveq2d 7430	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (1...(♯‘𝑤)) = (1...𝐾))
63	62	raleqdv 3319	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 → (𝑣‘𝑥) < (𝑣‘𝑦)) ↔ ∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑣‘𝑥) < (𝑣‘𝑦))))
64	63	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) ∧ 𝑥 ∈ (1...(♯‘𝑤))) → (∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 → (𝑣‘𝑥) < (𝑣‘𝑦)) ↔ ∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑣‘𝑥) < (𝑣‘𝑦))))
65	62, 64	raleqbidva 3325	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (∀𝑥 ∈ (1...(♯‘𝑤))∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 → (𝑣‘𝑥) < (𝑣‘𝑦)) ↔ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑣‘𝑥) < (𝑣‘𝑦))))
66	59, 65	mpbid 235	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑣‘𝑥) < (𝑣‘𝑦)))
67	53, 66	jca 521	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑣‘𝑥) < (𝑣‘𝑦))))
68		feq1 6681	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑣 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑣:(1...𝐾)⟶(1...𝑁)))
69		fveq1 6878	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑣 → (𝑓‘𝑥) = (𝑣‘𝑥))
70		fveq1 6878	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑣 → (𝑓‘𝑦) = (𝑣‘𝑦))
71	69, 70	breq12d 5116	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑣 → ((𝑓‘𝑥) < (𝑓‘𝑦) ↔ (𝑣‘𝑥) < (𝑣‘𝑦)))
72	71	imbi2d 343	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑣 → ((𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ (𝑥 < 𝑦 → (𝑣‘𝑥) < (𝑣‘𝑦))))
73	72	2ralbidv 3226	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑣 → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑣‘𝑥) < (𝑣‘𝑦))))
74	68, 73	anbi12d 644	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑣 → ((𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦))) ↔ (𝑣:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑣‘𝑥) < (𝑣‘𝑦)))))
75	50, 67, 74	elabd 3635	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣 ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))})
76	4	eleq2i 2852	. . . . . . . . . . 11 ⊢ (𝑣 ∈ 𝐴 ↔ 𝑣 ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))})
77	75, 76	sylibr 237	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣 ∈ 𝐴)
78	5	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝐹 = (𝑧 ∈ 𝐴 ↦ ran 𝑧))
79		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑧 = 𝑣) → 𝑧 = 𝑣)
80	79	rneqd 5922	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑧 = 𝑣) → ran 𝑧 = ran 𝑣)
81		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ 𝐴)
82		rnexg 7900	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ 𝐴 → ran 𝑣 ∈ V)
83	81, 82	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ran 𝑣 ∈ V)
84	78, 80, 81, 83	fvmptd 6995	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) = ran 𝑣)
85	84	ex 418	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑣 ∈ 𝐴 → (𝐹‘𝑣) = ran 𝑣))
86	85	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣 ∈ 𝐴 → (𝐹‘𝑣) = ran 𝑣))
87	77, 86	mpd 16	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝐹‘𝑣) = ran 𝑣)
88		dff1o2 6824	. . . . . . . . . . . . . . 15 ⊢ (𝑣:(1...𝐾)–1-1-onto→𝑤 ↔ (𝑣 Fn (1...𝐾) ∧ Fun ^◡𝑣 ∧ ran 𝑣 = 𝑤))
89	88	biimpi 219	. . . . . . . . . . . . . 14 ⊢ (𝑣:(1...𝐾)–1-1-onto→𝑤 → (𝑣 Fn (1...𝐾) ∧ Fun ^◡𝑣 ∧ ran 𝑣 = 𝑤))
90	89	simp3d 1162	. . . . . . . . . . . . 13 ⊢ (𝑣:(1...𝐾)–1-1-onto→𝑤 → ran 𝑣 = 𝑤)
91	45, 90	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → ran 𝑣 = 𝑤)
92	87, 91	eqtrd 2795	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝐹‘𝑣) = 𝑤)
93	92	eqcomd 2766	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑤 = (𝐹‘𝑣))
94	77, 93	jca 521	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣 ∈ 𝐴 ∧ 𝑤 = (𝐹‘𝑣)))
95	94	3expa 1136	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣 ∈ 𝐴 ∧ 𝑤 = (𝐹‘𝑣)))
96	95	ex 418	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤) → (𝑣 ∈ 𝐴 ∧ 𝑤 = (𝐹‘𝑣))))
97	96	eximdv 1950	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (∃𝑣 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤) → ∃𝑣(𝑣 ∈ 𝐴 ∧ 𝑤 = (𝐹‘𝑣))))
98	34, 97	mpd 16	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → ∃𝑣(𝑣 ∈ 𝐴 ∧ 𝑤 = (𝐹‘𝑣)))
99		df-rex 3087	. . . . 5 ⊢ (∃𝑣 ∈ 𝐴 𝑤 = (𝐹‘𝑣) ↔ ∃𝑣(𝑣 ∈ 𝐴 ∧ 𝑤 = (𝐹‘𝑣)))
100	98, 99	sylibr 237	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → ∃𝑣 ∈ 𝐴 𝑤 = (𝐹‘𝑣))
101	100	ralrimiva 3154	. . 3 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ∃𝑣 ∈ 𝐴 𝑤 = (𝐹‘𝑣))
102	10, 101	jca 521	. 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ∧ ∀𝑤 ∈ 𝐵 ∃𝑣 ∈ 𝐴 𝑤 = (𝐹‘𝑣)))
103		dffo3 7096	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑤 ∈ 𝐵 ∃𝑣 ∈ 𝐴 𝑤 = (𝐹‘𝑣)))
104	103	a1i 11	. 2 ⊢ (𝜑 → (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑤 ∈ 𝐵 ∃𝑣 ∈ 𝐴 𝑤 = (𝐹‘𝑣))))
105	102, 104	mpbird 260	1 ⊢ (𝜑 → 𝐹:𝐴–onto→𝐵)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∃wex 1812 ∈ wcel 2145 {cab 2738 ∀wral 3076 ∃wrex 3086 {crab 3412 Vcvv 3450 ⊆ wss 3899 𝒫 cpw 4557 class class class wbr 5103 ↦ cmpt 5186 Or wor 5562 ^◡ccnv 5654 ran crn 5656 Fun wfun 6527 Fn wfn 6528 ⟶wf 6529 –1-1→wf1 6530 –onto→wfo 6531 –1-1-onto→wf1o 6532 ‘cfv 6533 Isom wiso 6534 (class class class)co 7414 Fincfn 8955 ℝcr 11126 1c1 11128 < clt 11270 ℕcn 12260 ℕ₀cn0 12531 ...cfz 13564 ♯chash 14397

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 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205

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-se 5609 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-isom 6542 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 8458 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9415 df-inf 9416 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-n0 12532 df-z 12619 df-uz 12891 df-fz 13565 df-hash 14398

This theorem is used by: sticksstones4 43018

W3C validator