Users' Mathboxes Mathbox for metakunt < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sticksstones3 Structured version   Visualization version   GIF version

Theorem sticksstones3 41271
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 (𝜑𝑁 ∈ ℕ0)
sticksstones3.2 (𝜑𝐾 ∈ 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 → (𝑓𝑥) < (𝑓𝑦)))}
sticksstones3.5 𝐹 = (𝑧𝐴 ↦ ran 𝑧)
Assertion
Ref Expression
sticksstones3 (𝜑𝐹:𝐴onto𝐵)
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.
StepHypRef Expression
1 sticksstones3.1 . . . . 5 (𝜑𝑁 ∈ ℕ0)
2 sticksstones3.2 . . . . 5 (𝜑𝐾 ∈ 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 → (𝑓𝑥) < (𝑓𝑦)))}
5 sticksstones3.5 . . . . 5 𝐹 = (𝑧𝐴 ↦ ran 𝑧)
61, 2, 3, 4, 5sticksstones2 41270 . . . 4 (𝜑𝐹:𝐴1-1𝐵)
7 df-f1 6548 . . . . . 6 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
87biimpi 215 . . . . 5 (𝐹:𝐴1-1𝐵 → (𝐹:𝐴𝐵 ∧ Fun 𝐹))
98simpld 494 . . . 4 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
106, 9syl 17 . . 3 (𝜑𝐹:𝐴𝐵)
113eleq2i 2824 . . . . . . . . . . . . . . . . . . . . 21 (𝑤𝐵𝑤 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾})
1211biimpi 215 . . . . . . . . . . . . . . . . . . . 20 (𝑤𝐵𝑤 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾})
1312adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤𝐵) → 𝑤 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾})
14 fveqeq2 6900 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑤 → ((♯‘𝑎) = 𝐾 ↔ (♯‘𝑤) = 𝐾))
1514elrab 3683 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾} ↔ (𝑤 ∈ 𝒫 (1...𝑁) ∧ (♯‘𝑤) = 𝐾))
1613, 15sylib 217 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤𝐵) → (𝑤 ∈ 𝒫 (1...𝑁) ∧ (♯‘𝑤) = 𝐾))
1716simpld 494 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝐵) → 𝑤 ∈ 𝒫 (1...𝑁))
1817elpwid 4611 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝐵) → 𝑤 ⊆ (1...𝑁))
1918sseld 3981 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝐵) → (𝑐𝑤𝑐 ∈ (1...𝑁)))
2019imp 406 . . . . . . . . . . . . . 14 (((𝜑𝑤𝐵) ∧ 𝑐𝑤) → 𝑐 ∈ (1...𝑁))
21203impa 1109 . . . . . . . . . . . . 13 ((𝜑𝑤𝐵𝑐𝑤) → 𝑐 ∈ (1...𝑁))
22 elfznn 13535 . . . . . . . . . . . . 13 (𝑐 ∈ (1...𝑁) → 𝑐 ∈ ℕ)
2321, 22syl 17 . . . . . . . . . . . 12 ((𝜑𝑤𝐵𝑐𝑤) → 𝑐 ∈ ℕ)
2423nnred 12232 . . . . . . . . . . 11 ((𝜑𝑤𝐵𝑐𝑤) → 𝑐 ∈ ℝ)
25243expa 1117 . . . . . . . . . 10 (((𝜑𝑤𝐵) ∧ 𝑐𝑤) → 𝑐 ∈ ℝ)
2625ex 412 . . . . . . . . 9 ((𝜑𝑤𝐵) → (𝑐𝑤𝑐 ∈ ℝ))
2726ssrdv 3988 . . . . . . . 8 ((𝜑𝑤𝐵) → 𝑤 ⊆ ℝ)
28 ltso 11299 . . . . . . . . 9 < Or ℝ
29 soss 5608 . . . . . . . . 9 (𝑤 ⊆ ℝ → ( < Or ℝ → < Or 𝑤))
3028, 29mpi 20 . . . . . . . 8 (𝑤 ⊆ ℝ → < Or 𝑤)
3127, 30syl 17 . . . . . . 7 ((𝜑𝑤𝐵) → < Or 𝑤)
32 fzfid 13943 . . . . . . . 8 ((𝜑𝑤𝐵) → (1...𝑁) ∈ Fin)
3332, 18ssfid 9271 . . . . . . 7 ((𝜑𝑤𝐵) → 𝑤 ∈ Fin)
34 fz1iso 14428 . . . . . . 7 (( < Or 𝑤𝑤 ∈ Fin) → ∃𝑣 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤))
3531, 33, 34syl2anc 583 . . . . . 6 ((𝜑𝑤𝐵) → ∃𝑣 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤))
36 df-isom 6552 . . . . . . . . . . . . . . . . . . 19 (𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤) ↔ (𝑣:(1...(♯‘𝑤))–1-1-onto𝑤 ∧ ∀𝑥 ∈ (1...(♯‘𝑤))∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 ↔ (𝑣𝑥) < (𝑣𝑦))))
3736biimpi 215 . . . . . . . . . . . . . . . . . 18 (𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤) → (𝑣:(1...(♯‘𝑤))–1-1-onto𝑤 ∧ ∀𝑥 ∈ (1...(♯‘𝑤))∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 ↔ (𝑣𝑥) < (𝑣𝑦))))
38373ad2ant3 1134 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣:(1...(♯‘𝑤))–1-1-onto𝑤 ∧ ∀𝑥 ∈ (1...(♯‘𝑤))∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 ↔ (𝑣𝑥) < (𝑣𝑦))))
3938simpld 494 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣:(1...(♯‘𝑤))–1-1-onto𝑤)
4016simprd 495 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤𝐵) → (♯‘𝑤) = 𝐾)
41 oveq2 7420 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑤) = 𝐾 → (1...(♯‘𝑤)) = (1...𝐾))
4241f1oeq2d 6829 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝑤) = 𝐾 → (𝑣:(1...(♯‘𝑤))–1-1-onto𝑤𝑣:(1...𝐾)–1-1-onto𝑤))
4340, 42syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤𝐵) → (𝑣:(1...(♯‘𝑤))–1-1-onto𝑤𝑣:(1...𝐾)–1-1-onto𝑤))
4443biimpd 228 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝐵) → (𝑣:(1...(♯‘𝑤))–1-1-onto𝑤𝑣:(1...𝐾)–1-1-onto𝑤))
45443adant3 1131 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣:(1...(♯‘𝑤))–1-1-onto𝑤𝑣:(1...𝐾)–1-1-onto𝑤))
4639, 45mpd 15 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣:(1...𝐾)–1-1-onto𝑤)
47 f1of 6833 . . . . . . . . . . . . . . 15 (𝑣:(1...𝐾)–1-1-onto𝑤𝑣:(1...𝐾)⟶𝑤)
4846, 47syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣:(1...𝐾)⟶𝑤)
4948ffnd 6718 . . . . . . . . . . . . 13 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣 Fn (1...𝐾))
50 ovexd 7447 . . . . . . . . . . . . 13 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (1...𝐾) ∈ V)
5149, 50fnexd 7222 . . . . . . . . . . . 12 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣 ∈ V)
52183adant3 1131 . . . . . . . . . . . . . 14 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑤 ⊆ (1...𝑁))
53 fss 6734 . . . . . . . . . . . . . 14 ((𝑣:(1...𝐾)⟶𝑤𝑤 ⊆ (1...𝑁)) → 𝑣:(1...𝐾)⟶(1...𝑁))
5448, 52, 53syl2anc 583 . . . . . . . . . . . . 13 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣:(1...𝐾)⟶(1...𝑁))
5538simprd 495 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → ∀𝑥 ∈ (1...(♯‘𝑤))∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 ↔ (𝑣𝑥) < (𝑣𝑦)))
56 biimp 214 . . . . . . . . . . . . . . . . . 18 ((𝑥 < 𝑦 ↔ (𝑣𝑥) < (𝑣𝑦)) → (𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦)))
5756a1i 11 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) ∧ 𝑥 ∈ (1...(♯‘𝑤))) ∧ 𝑦 ∈ (1...(♯‘𝑤))) → ((𝑥 < 𝑦 ↔ (𝑣𝑥) < (𝑣𝑦)) → (𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦))))
5857ralimdva 3166 . . . . . . . . . . . . . . . 16 (((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) ∧ 𝑥 ∈ (1...(♯‘𝑤))) → (∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 ↔ (𝑣𝑥) < (𝑣𝑦)) → ∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦))))
5958ralimdva 3166 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (∀𝑥 ∈ (1...(♯‘𝑤))∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 ↔ (𝑣𝑥) < (𝑣𝑦)) → ∀𝑥 ∈ (1...(♯‘𝑤))∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦))))
6055, 59mpd 15 . . . . . . . . . . . . . 14 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → ∀𝑥 ∈ (1...(♯‘𝑤))∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦)))
6140adantr 480 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤𝐵) ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (♯‘𝑤) = 𝐾)
62613impa 1109 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (♯‘𝑤) = 𝐾)
6362oveq2d 7428 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (1...(♯‘𝑤)) = (1...𝐾))
6463raleqdv 3324 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦)) ↔ ∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦))))
6564adantr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) ∧ 𝑥 ∈ (1...(♯‘𝑤))) → (∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦)) ↔ ∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦))))
6663, 65raleqbidva 3326 . . . . . . . . . . . . . 14 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (∀𝑥 ∈ (1...(♯‘𝑤))∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦)) ↔ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦))))
6760, 66mpbid 231 . . . . . . . . . . . . 13 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦)))
6854, 67jca 511 . . . . . . . . . . . 12 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦))))
69 feq1 6698 . . . . . . . . . . . . 13 (𝑓 = 𝑣 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑣:(1...𝐾)⟶(1...𝑁)))
70 fveq1 6890 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑣 → (𝑓𝑥) = (𝑣𝑥))
71 fveq1 6890 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑣 → (𝑓𝑦) = (𝑣𝑦))
7270, 71breq12d 5161 . . . . . . . . . . . . . . 15 (𝑓 = 𝑣 → ((𝑓𝑥) < (𝑓𝑦) ↔ (𝑣𝑥) < (𝑣𝑦)))
7372imbi2d 340 . . . . . . . . . . . . . 14 (𝑓 = 𝑣 → ((𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)) ↔ (𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦))))
74732ralbidv 3217 . . . . . . . . . . . . 13 (𝑓 = 𝑣 → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)) ↔ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦))))
7569, 74anbi12d 630 . . . . . . . . . . . 12 (𝑓 = 𝑣 → ((𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦))) ↔ (𝑣:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦)))))
7651, 68, 75elabd 3671 . . . . . . . . . . 11 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣 ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))})
774eleq2i 2824 . . . . . . . . . . 11 (𝑣𝐴𝑣 ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))})
7876, 77sylibr 233 . . . . . . . . . 10 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣𝐴)
795a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑣𝐴) → 𝐹 = (𝑧𝐴 ↦ ran 𝑧))
80 simpr 484 . . . . . . . . . . . . . . . . 17 (((𝜑𝑣𝐴) ∧ 𝑧 = 𝑣) → 𝑧 = 𝑣)
8180rneqd 5937 . . . . . . . . . . . . . . . 16 (((𝜑𝑣𝐴) ∧ 𝑧 = 𝑣) → ran 𝑧 = ran 𝑣)
82 simpr 484 . . . . . . . . . . . . . . . 16 ((𝜑𝑣𝐴) → 𝑣𝐴)
83 rnexg 7899 . . . . . . . . . . . . . . . . 17 (𝑣𝐴 → ran 𝑣 ∈ V)
8482, 83syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑣𝐴) → ran 𝑣 ∈ V)
8579, 81, 82, 84fvmptd 7005 . . . . . . . . . . . . . . 15 ((𝜑𝑣𝐴) → (𝐹𝑣) = ran 𝑣)
8685ex 412 . . . . . . . . . . . . . 14 (𝜑 → (𝑣𝐴 → (𝐹𝑣) = ran 𝑣))
87863ad2ant1 1132 . . . . . . . . . . . . 13 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣𝐴 → (𝐹𝑣) = ran 𝑣))
8878, 87mpd 15 . . . . . . . . . . . 12 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝐹𝑣) = ran 𝑣)
89 dff1o2 6838 . . . . . . . . . . . . . . 15 (𝑣:(1...𝐾)–1-1-onto𝑤 ↔ (𝑣 Fn (1...𝐾) ∧ Fun 𝑣 ∧ ran 𝑣 = 𝑤))
9089biimpi 215 . . . . . . . . . . . . . 14 (𝑣:(1...𝐾)–1-1-onto𝑤 → (𝑣 Fn (1...𝐾) ∧ Fun 𝑣 ∧ ran 𝑣 = 𝑤))
9190simp3d 1143 . . . . . . . . . . . . 13 (𝑣:(1...𝐾)–1-1-onto𝑤 → ran 𝑣 = 𝑤)
9246, 91syl 17 . . . . . . . . . . . 12 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → ran 𝑣 = 𝑤)
9388, 92eqtrd 2771 . . . . . . . . . . 11 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝐹𝑣) = 𝑤)
9493eqcomd 2737 . . . . . . . . . 10 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑤 = (𝐹𝑣))
9578, 94jca 511 . . . . . . . . 9 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣𝐴𝑤 = (𝐹𝑣)))
96953expa 1117 . . . . . . . 8 (((𝜑𝑤𝐵) ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣𝐴𝑤 = (𝐹𝑣)))
9796ex 412 . . . . . . 7 ((𝜑𝑤𝐵) → (𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤) → (𝑣𝐴𝑤 = (𝐹𝑣))))
9897eximdv 1919 . . . . . 6 ((𝜑𝑤𝐵) → (∃𝑣 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤) → ∃𝑣(𝑣𝐴𝑤 = (𝐹𝑣))))
9935, 98mpd 15 . . . . 5 ((𝜑𝑤𝐵) → ∃𝑣(𝑣𝐴𝑤 = (𝐹𝑣)))
100 df-rex 3070 . . . . 5 (∃𝑣𝐴 𝑤 = (𝐹𝑣) ↔ ∃𝑣(𝑣𝐴𝑤 = (𝐹𝑣)))
10199, 100sylibr 233 . . . 4 ((𝜑𝑤𝐵) → ∃𝑣𝐴 𝑤 = (𝐹𝑣))
102101ralrimiva 3145 . . 3 (𝜑 → ∀𝑤𝐵𝑣𝐴 𝑤 = (𝐹𝑣))
10310, 102jca 511 . 2 (𝜑 → (𝐹:𝐴𝐵 ∧ ∀𝑤𝐵𝑣𝐴 𝑤 = (𝐹𝑣)))
104 dffo3 7103 . . 3 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑤𝐵𝑣𝐴 𝑤 = (𝐹𝑣)))
105104a1i 11 . 2 (𝜑 → (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑤𝐵𝑣𝐴 𝑤 = (𝐹𝑣))))
106103, 105mpbird 257 1 (𝜑𝐹:𝐴onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wex 1780  wcel 2105  {cab 2708  wral 3060  wrex 3069  {crab 3431  Vcvv 3473  wss 3948  𝒫 cpw 4602   class class class wbr 5148  cmpt 5231   Or wor 5587  ccnv 5675  ran crn 5677  Fun wfun 6537   Fn wfn 6538  wf 6539  1-1wf1 6540  ontowfo 6541  1-1-ontowf1o 6542  cfv 6543   Isom wiso 6544  (class class class)co 7412  Fincfn 8943  cr 11113  1c1 11115   < clt 11253  cn 12217  0cn0 12477  ...cfz 13489  chash 14295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191  ax-pre-sup 11192
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-sup 9441  df-inf 9442  df-oi 9509  df-card 9938  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-n0 12478  df-z 12564  df-uz 12828  df-fz 13490  df-hash 14296
This theorem is referenced by:  sticksstones4  41272
  Copyright terms: Public domain W3C validator