sticksstones3 - Mathbox for metakunt

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Theorem sticksstones3 42975

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 42974	. . . 4 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)
7		df-f1 6545	. . . . . 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 2857	. . . . . . . . . . . . . . . . . . . 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 6894	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑤 → ((♯‘𝑎) = 𝐾 ↔ (♯‘𝑤) = 𝐾))
14	13	elrab 3652	. . . . . . . . . . . . . . . . . . 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 4573	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ⊆ (1...𝑁))
18	17	sseld 3937	. . . . . . . . . . . . . . 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 13600	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (1...𝑁) → 𝑐 ∈ ℕ)
22	20, 21	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑐 ∈ 𝑤) → 𝑐 ∈ ℕ)
23	22	nnred 12265	. . . . . . . . . . 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 3944	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ⊆ ℝ)
27		ltso 11307	. . . . . . . . 9 ⊢ < Or ℝ
28		soss 5591	. . . . . . . . 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 14029	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (1...𝑁) ∈ Fin)
32	31, 17	ssfid 9236	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ Fin)
33		fz1iso 14519	. . . . . . 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 6549	. . . . . . . . . . . . . . . . . . 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 7427	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑤) = 𝐾 → (1...(♯‘𝑤)) = (1...𝐾))
41	40	f1oeq2d 6820	. . . . . . . . . . . . . . . . . . 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 6824	. . . . . . . . . . . . . . 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 6710	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣 Fn (1...𝐾))
49		ovexd 7454	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (1...𝐾) ∈ V)
50	48, 49	fnexd 7223	. . . . . . . . . . . 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 6726	. . . . . . . . . . . . . 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 3179	. . . . . . . . . . . . . . . 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 3179	. . . . . . . . . . . . . . 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 7435	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (1...(♯‘𝑤)) = (1...𝐾))
63	62	raleqdv 3325	. . . . . . . . . . . . . . . 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 3331	. . . . . . . . . . . . . 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 6687	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑣 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑣:(1...𝐾)⟶(1...𝑁)))
69		fveq1 6884	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑣 → (𝑓‘𝑥) = (𝑣‘𝑥))
70		fveq1 6884	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑣 → (𝑓‘𝑦) = (𝑣‘𝑦))
71	69, 70	breq12d 5124	. . . . . . . . . . . . . . 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 3231	. . . . . . . . . . . . 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 3642	. . . . . . . . . . 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 2857	. . . . . . . . . . 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 5930	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑧 = 𝑣) → ran 𝑧 = ran 𝑣)
81		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ 𝐴)
82		rnexg 7905	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ 𝐴 → ran 𝑣 ∈ V)
83	81, 82	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ran 𝑣 ∈ V)
84	78, 80, 81, 83	fvmptd 7001	. . . . . . . . . . . . . . 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 6830	. . . . . . . . . . . . . . 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 2800	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵 ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝐹‘𝑣) = 𝑤)
93	92	eqcomd 2771	. . . . . . . . . 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 3092	. . . . 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 3159	. . 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 7101	. . 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 2146 {cab 2743 ∀wral 3081 ∃wrex 3091 {crab 3418 Vcvv 3457 ⊆ wss 3906 𝒫 cpw 4564 class class class wbr 5111 ↦ cmpt 5194 Or wor 5570 ^◡ccnv 5662 ran crn 5664 Fun wfun 6534 Fn wfn 6535 ⟶wf 6536 –1-1→wf1 6537 –onto→wfo 6538 –1-1-onto→wf1o 6539 ‘cfv 6540 Isom wiso 6541 (class class class)co 7419 Fincfn 8949 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-se 5617 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-isom 6549 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-oi 9479 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: sticksstones4 42976

W3C validator