sticksstones3 - Mathbox for metakunt

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Theorem sticksstones3 40104

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 40103	. . . 4 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)
7		df-f1 6438	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ^◡𝐹))
8	7	biimpi 215	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹:𝐴⟶𝐵 ∧ Fun ^◡𝐹))
9	8	simpld 495	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
10	6, 9	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
11	3	eleq2i 2830	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ 𝐵 ↔ 𝑤 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾})
12	11	biimpi 215	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ 𝐵 → 𝑤 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾})
13	12	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾})
14		fveqeq2 6783	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑤 → ((♯‘𝑎) = 𝐾 ↔ (♯‘𝑤) = 𝐾))
15	14	elrab 3624	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾} ↔ (𝑤 ∈ 𝒫 (1...𝑁) ∧ (♯‘𝑤) = 𝐾))
16	13, 15	sylib 217	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝑤 ∈ 𝒫 (1...𝑁) ∧ (♯‘𝑤) = 𝐾))
17	16	simpld 495	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ 𝒫 (1...𝑁))
18	17	elpwid 4544	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ⊆ (1...𝑁))
19	18	sseld 3920	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝑐 ∈ 𝑤 → 𝑐 ∈ (1...𝑁)))
20	19	imp 407	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ 𝑐 ∈ 𝑤) → 𝑐 ∈ (1...𝑁))
21	20	3impa 1109	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑐 ∈ 𝑤) → 𝑐 ∈ (1...𝑁))
22		elfznn 13285	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (1...𝑁) → 𝑐 ∈ ℕ)
23	21, 22	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑐 ∈ 𝑤) → 𝑐 ∈ ℕ)
24	23	nnred 11988	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑐 ∈ 𝑤) → 𝑐 ∈ ℝ)
25	24	3expa 1117	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ 𝑐 ∈ 𝑤) → 𝑐 ∈ ℝ)
26	25	ex 413	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝑐 ∈ 𝑤 → 𝑐 ∈ ℝ))
27	26	ssrdv 3927	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ⊆ ℝ)
28		ltso 11055	. . . . . . . . 9 ⊢ < Or ℝ
29		soss 5523	. . . . . . . . 9 ⊢ (𝑤 ⊆ ℝ → ( < Or ℝ → < Or 𝑤))
30	28, 29	mpi 20	. . . . . . . 8 ⊢ (𝑤 ⊆ ℝ → < Or 𝑤)
31	27, 30	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → < Or 𝑤)
32		fzfid 13693	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (1...𝑁) ∈ Fin)
33	32, 18	ssfid 9042	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ Fin)
34		fz1iso 14176	. . . . . . 7 ⊢ (( < Or 𝑤 ∧ 𝑤 ∈ Fin) → ∃𝑣 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤))
35	31, 33, 34	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → ∃𝑣 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤))
36		df-isom 6442	. . . . . . . . . . . . . . . . . . 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))))
37	36	biimpi 215	. . . . . . . . . . . . . . . . . 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))))
38	37	3ad2ant3 1134	. . . . . . . . . . . . . . . . 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))))
39	38	simpld 495	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣:(1...(♯‘𝑤))–1-1-onto→𝑤)
40	16	simprd 496	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (♯‘𝑤) = 𝐾)
41		oveq2 7283	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑤) = 𝐾 → (1...(♯‘𝑤)) = (1...𝐾))
42	41	f1oeq2d 6712	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝑤) = 𝐾 → (𝑣:(1...(♯‘𝑤))–1-1-onto→𝑤 ↔ 𝑣:(1...𝐾)–1-1-onto→𝑤))
43	40, 42	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝑣:(1...(♯‘𝑤))–1-1-onto→𝑤 ↔ 𝑣:(1...𝐾)–1-1-onto→𝑤))
44	43	biimpd 228	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝑣:(1...(♯‘𝑤))–1-1-onto→𝑤 → 𝑣:(1...𝐾)–1-1-onto→𝑤))
45	44	3adant3 1131	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣:(1...(♯‘𝑤))–1-1-onto→𝑤 → 𝑣:(1...𝐾)–1-1-onto→𝑤))
46	39, 45	mpd 15	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣:(1...𝐾)–1-1-onto→𝑤)
47		f1of 6716	. . . . . . . . . . . . . . 15 ⊢ (𝑣:(1...𝐾)–1-1-onto→𝑤 → 𝑣:(1...𝐾)⟶𝑤)
48	46, 47	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣:(1...𝐾)⟶𝑤)
49	48	ffnd 6601	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣 Fn (1...𝐾))
50		ovexd 7310	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (1...𝐾) ∈ V)
51	49, 50	fnexd 7094	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣 ∈ V)
52	18	3adant3 1131	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑤 ⊆ (1...𝑁))
53		fss 6617	. . . . . . . . . . . . . 14 ⊢ ((𝑣:(1...𝐾)⟶𝑤 ∧ 𝑤 ⊆ (1...𝑁)) → 𝑣:(1...𝐾)⟶(1...𝑁))
54	48, 52, 53	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣:(1...𝐾)⟶(1...𝑁))
55	38	simprd 496	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → ∀𝑥 ∈ (1...(♯‘𝑤))∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 ↔ (𝑣‘𝑥) < (𝑣‘𝑦)))
56		biimp 214	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 < 𝑦 ↔ (𝑣‘𝑥) < (𝑣‘𝑦)) → (𝑥 < 𝑦 → (𝑣‘𝑥) < (𝑣‘𝑦)))
57	56	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))))
58	57	ralimdva 3108	. . . . . . . . . . . . . . . 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))))
59	58	ralimdva 3108	. . . . . . . . . . . . . . 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))))
60	55, 59	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → ∀𝑥 ∈ (1...(♯‘𝑤))∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 → (𝑣‘𝑥) < (𝑣‘𝑦)))
61	40	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (♯‘𝑤) = 𝐾)
62	61	3impa 1109	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (♯‘𝑤) = 𝐾)
63	62	oveq2d 7291	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (1...(♯‘𝑤)) = (1...𝐾))
64	63	raleqdv 3348	. . . . . . . . . . . . . . . 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))))
65	64	adantr 481	. . . . . . . . . . . . . . 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))))
66	63, 65	raleqbidva 3354	. . . . . . . . . . . . . 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))))
67	60, 66	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑣‘𝑥) < (𝑣‘𝑦)))
68	54, 67	jca 512	. . . . . . . . . . . 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))))
69		feq1 6581	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑣 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑣:(1...𝐾)⟶(1...𝑁)))
70		fveq1 6773	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑣 → (𝑓‘𝑥) = (𝑣‘𝑥))
71		fveq1 6773	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑣 → (𝑓‘𝑦) = (𝑣‘𝑦))
72	70, 71	breq12d 5087	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑣 → ((𝑓‘𝑥) < (𝑓‘𝑦) ↔ (𝑣‘𝑥) < (𝑣‘𝑦)))
73	72	imbi2d 341	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑣 → ((𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ (𝑥 < 𝑦 → (𝑣‘𝑥) < (𝑣‘𝑦))))
74	73	2ralbidv 3129	. . . . . . . . . . . . 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))))
75	69, 74	anbi12d 631	. . . . . . . . . . . 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)))))
76	51, 68, 75	elabd 3612	. . . . . . . . . . 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)))})
77	4	eleq2i 2830	. . . . . . . . . . 11 ⊢ (𝑣 ∈ 𝐴 ↔ 𝑣 ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))})
78	76, 77	sylibr 233	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣 ∈ 𝐴)
79	5	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝐹 = (𝑧 ∈ 𝐴 ↦ ran 𝑧))
80		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑧 = 𝑣) → 𝑧 = 𝑣)
81	80	rneqd 5847	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑧 = 𝑣) → ran 𝑧 = ran 𝑣)
82		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ 𝐴)
83		rnexg 7751	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ 𝐴 → ran 𝑣 ∈ V)
84	82, 83	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ran 𝑣 ∈ V)
85	79, 81, 82, 84	fvmptd 6882	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) = ran 𝑣)
86	85	ex 413	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑣 ∈ 𝐴 → (𝐹‘𝑣) = ran 𝑣))
87	86	3ad2ant1 1132	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣 ∈ 𝐴 → (𝐹‘𝑣) = ran 𝑣))
88	78, 87	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝐹‘𝑣) = ran 𝑣)
89		dff1o2 6721	. . . . . . . . . . . . . . 15 ⊢ (𝑣:(1...𝐾)–1-1-onto→𝑤 ↔ (𝑣 Fn (1...𝐾) ∧ Fun ^◡𝑣 ∧ ran 𝑣 = 𝑤))
90	89	biimpi 215	. . . . . . . . . . . . . 14 ⊢ (𝑣:(1...𝐾)–1-1-onto→𝑤 → (𝑣 Fn (1...𝐾) ∧ Fun ^◡𝑣 ∧ ran 𝑣 = 𝑤))
91	90	simp3d 1143	. . . . . . . . . . . . 13 ⊢ (𝑣:(1...𝐾)–1-1-onto→𝑤 → ran 𝑣 = 𝑤)
92	46, 91	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → ran 𝑣 = 𝑤)
93	88, 92	eqtrd 2778	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝐹‘𝑣) = 𝑤)
94	93	eqcomd 2744	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑤 = (𝐹‘𝑣))
95	78, 94	jca 512	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣 ∈ 𝐴 ∧ 𝑤 = (𝐹‘𝑣)))
96	95	3expa 1117	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣 ∈ 𝐴 ∧ 𝑤 = (𝐹‘𝑣)))
97	96	ex 413	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤) → (𝑣 ∈ 𝐴 ∧ 𝑤 = (𝐹‘𝑣))))
98	97	eximdv 1920	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (∃𝑣 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤) → ∃𝑣(𝑣 ∈ 𝐴 ∧ 𝑤 = (𝐹‘𝑣))))
99	35, 98	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → ∃𝑣(𝑣 ∈ 𝐴 ∧ 𝑤 = (𝐹‘𝑣)))
100		df-rex 3070	. . . . 5 ⊢ (∃𝑣 ∈ 𝐴 𝑤 = (𝐹‘𝑣) ↔ ∃𝑣(𝑣 ∈ 𝐴 ∧ 𝑤 = (𝐹‘𝑣)))
101	99, 100	sylibr 233	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → ∃𝑣 ∈ 𝐴 𝑤 = (𝐹‘𝑣))
102	101	ralrimiva 3103	. . 3 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ∃𝑣 ∈ 𝐴 𝑤 = (𝐹‘𝑣))
103	10, 102	jca 512	. 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ∧ ∀𝑤 ∈ 𝐵 ∃𝑣 ∈ 𝐴 𝑤 = (𝐹‘𝑣)))
104		dffo3 6978	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑤 ∈ 𝐵 ∃𝑣 ∈ 𝐴 𝑤 = (𝐹‘𝑣)))
105	104	a1i 11	. 2 ⊢ (𝜑 → (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑤 ∈ 𝐵 ∃𝑣 ∈ 𝐴 𝑤 = (𝐹‘𝑣))))
106	103, 105	mpbird 256	1 ⊢ (𝜑 → 𝐹:𝐴–onto→𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∃wex 1782 ∈ wcel 2106 {cab 2715 ∀wral 3064 ∃wrex 3065 {crab 3068 Vcvv 3432 ⊆ wss 3887 𝒫 cpw 4533 class class class wbr 5074 ↦ cmpt 5157 Or wor 5502 ^◡ccnv 5588 ran crn 5590 Fun wfun 6427 Fn wfn 6428 ⟶wf 6429 –1-1→wf1 6430 –onto→wfo 6431 –1-1-onto→wf1o 6432 ‘cfv 6433 Isom wiso 6434 (class class class)co 7275 Fincfn 8733 ℝcr 10870 1c1 10872 < clt 11009 ℕcn 11973 ℕ₀cn0 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-se 5545 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-isom 6442 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-oi 9269 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: sticksstones4 40105

W3C validator