sticksstones3 - Mathbox for metakunt

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Theorem sticksstones3 40769

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 40768	. . . 4 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)
7		df-f1 6537	. . . . . 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 2824	. . . . . . . . . . . . . . . . . . . . 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 6887	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑤 → ((♯‘𝑎) = 𝐾 ↔ (♯‘𝑤) = 𝐾))
15	14	elrab 3679	. . . . . . . . . . . . . . . . . . 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 4605	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ⊆ (1...𝑁))
19	18	sseld 3977	. . . . . . . . . . . . . . 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 1110	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑐 ∈ 𝑤) → 𝑐 ∈ (1...𝑁))
22		elfznn 13512	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (1...𝑁) → 𝑐 ∈ ℕ)
23	21, 22	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑐 ∈ 𝑤) → 𝑐 ∈ ℕ)
24	23	nnred 12209	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑐 ∈ 𝑤) → 𝑐 ∈ ℝ)
25	24	3expa 1118	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ 𝑐 ∈ 𝑤) → 𝑐 ∈ ℝ)
26	25	ex 413	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝑐 ∈ 𝑤 → 𝑐 ∈ ℝ))
27	26	ssrdv 3984	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ⊆ ℝ)
28		ltso 11276	. . . . . . . . 9 ⊢ < Or ℝ
29		soss 5601	. . . . . . . . 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 13920	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (1...𝑁) ∈ Fin)
33	32, 18	ssfid 9250	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ Fin)
34		fz1iso 14405	. . . . . . 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 6541	. . . . . . . . . . . . . . . . . . 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 1135	. . . . . . . . . . . . . . . . 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 7401	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑤) = 𝐾 → (1...(♯‘𝑤)) = (1...𝐾))
42	41	f1oeq2d 6816	. . . . . . . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . . . . 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 6820	. . . . . . . . . . . . . . 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 6705	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣 Fn (1...𝐾))
50		ovexd 7428	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (1...𝐾) ∈ V)
51	49, 50	fnexd 7204	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣 ∈ V)
52	18	3adant3 1132	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑤 ⊆ (1...𝑁))
53		fss 6721	. . . . . . . . . . . . . 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 3166	. . . . . . . . . . . . . . . 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 3166	. . . . . . . . . . . . . . 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 1110	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (♯‘𝑤) = 𝐾)
63	62	oveq2d 7409	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (1...(♯‘𝑤)) = (1...𝐾))
64	63	raleqdv 3324	. . . . . . . . . . . . . . . 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 3326	. . . . . . . . . . . . . 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 6685	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑣 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑣:(1...𝐾)⟶(1...𝑁)))
70		fveq1 6877	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑣 → (𝑓‘𝑥) = (𝑣‘𝑥))
71		fveq1 6877	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑣 → (𝑓‘𝑦) = (𝑣‘𝑦))
72	70, 71	breq12d 5154	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑣 → ((𝑓‘𝑥) < (𝑓‘𝑦) ↔ (𝑣‘𝑥) < (𝑣‘𝑦)))
73	72	imbi2d 340	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑣 → ((𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ (𝑥 < 𝑦 → (𝑣‘𝑥) < (𝑣‘𝑦))))
74	73	2ralbidv 3217	. . . . . . . . . . . . 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 3667	. . . . . . . . . . 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 2824	. . . . . . . . . . 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 5929	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑧 = 𝑣) → ran 𝑧 = ran 𝑣)
82		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ 𝐴)
83		rnexg 7877	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ 𝐴 → ran 𝑣 ∈ V)
84	82, 83	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ran 𝑣 ∈ V)
85	79, 81, 82, 84	fvmptd 6991	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) = ran 𝑣)
86	85	ex 413	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑣 ∈ 𝐴 → (𝐹‘𝑣) = ran 𝑣))
87	86	3ad2ant1 1133	. . . . . . . . . . . . 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 6825	. . . . . . . . . . . . . . 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 1144	. . . . . . . . . . . . 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 2771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝐹‘𝑣) = 𝑤)
94	93	eqcomd 2737	. . . . . . . . . 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 1118	. . . . . . . 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 3145	. . 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 7088	. . 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 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 {cab 2708 ∀wral 3060 ∃wrex 3069 {crab 3431 Vcvv 3473 ⊆ wss 3944 𝒫 cpw 4596 class class class wbr 5141 ↦ cmpt 5224 Or wor 5580 ^◡ccnv 5668 ran crn 5670 Fun wfun 6526 Fn wfn 6527 ⟶wf 6528 –1-1→wf1 6529 –onto→wfo 6530 –1-1-onto→wf1o 6531 ‘cfv 6532 Isom wiso 6533 (class class class)co 7393 Fincfn 8922 ℝcr 11091 1c1 11093 < clt 11230 ℕcn 12194 ℕ₀cn0 12454 ...cfz 13466 ♯chash 14272

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-isom 6541 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-sup 9419 df-inf 9420 df-oi 9487 df-card 9916 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-n0 12455 df-z 12541 df-uz 12805 df-fz 13467 df-hash 14273

This theorem is referenced by: sticksstones4 40770

W3C validator