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 40556
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 40555 . . . 4 (𝜑𝐹:𝐴1-1𝐵)
7 df-f1 6501 . . . . . 6 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
87biimpi 215 . . . . 5 (𝐹:𝐴1-1𝐵 → (𝐹:𝐴𝐵 ∧ Fun 𝐹))
98simpld 495 . . . 4 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
106, 9syl 17 . . 3 (𝜑𝐹:𝐴𝐵)
113eleq2i 2829 . . . . . . . . . . . . . . . . . . . . 21 (𝑤𝐵𝑤 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾})
1211biimpi 215 . . . . . . . . . . . . . . . . . . . 20 (𝑤𝐵𝑤 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾})
1312adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤𝐵) → 𝑤 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾})
14 fveqeq2 6851 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑤 → ((♯‘𝑎) = 𝐾 ↔ (♯‘𝑤) = 𝐾))
1514elrab 3645 . . . . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝐵) → 𝑤 ∈ 𝒫 (1...𝑁))
1817elpwid 4569 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝐵) → 𝑤 ⊆ (1...𝑁))
1918sseld 3943 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝐵) → (𝑐𝑤𝑐 ∈ (1...𝑁)))
2019imp 407 . . . . . . . . . . . . . 14 (((𝜑𝑤𝐵) ∧ 𝑐𝑤) → 𝑐 ∈ (1...𝑁))
21203impa 1110 . . . . . . . . . . . . 13 ((𝜑𝑤𝐵𝑐𝑤) → 𝑐 ∈ (1...𝑁))
22 elfznn 13470 . . . . . . . . . . . . 13 (𝑐 ∈ (1...𝑁) → 𝑐 ∈ ℕ)
2321, 22syl 17 . . . . . . . . . . . 12 ((𝜑𝑤𝐵𝑐𝑤) → 𝑐 ∈ ℕ)
2423nnred 12168 . . . . . . . . . . 11 ((𝜑𝑤𝐵𝑐𝑤) → 𝑐 ∈ ℝ)
25243expa 1118 . . . . . . . . . 10 (((𝜑𝑤𝐵) ∧ 𝑐𝑤) → 𝑐 ∈ ℝ)
2625ex 413 . . . . . . . . 9 ((𝜑𝑤𝐵) → (𝑐𝑤𝑐 ∈ ℝ))
2726ssrdv 3950 . . . . . . . 8 ((𝜑𝑤𝐵) → 𝑤 ⊆ ℝ)
28 ltso 11235 . . . . . . . . 9 < Or ℝ
29 soss 5565 . . . . . . . . 9 (𝑤 ⊆ ℝ → ( < Or ℝ → < Or 𝑤))
3028, 29mpi 20 . . . . . . . 8 (𝑤 ⊆ ℝ → < Or 𝑤)
3127, 30syl 17 . . . . . . 7 ((𝜑𝑤𝐵) → < Or 𝑤)
32 fzfid 13878 . . . . . . . 8 ((𝜑𝑤𝐵) → (1...𝑁) ∈ Fin)
3332, 18ssfid 9211 . . . . . . 7 ((𝜑𝑤𝐵) → 𝑤 ∈ Fin)
34 fz1iso 14361 . . . . . . 7 (( < Or 𝑤𝑤 ∈ Fin) → ∃𝑣 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤))
3531, 33, 34syl2anc 584 . . . . . 6 ((𝜑𝑤𝐵) → ∃𝑣 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤))
36 df-isom 6505 . . . . . . . . . . . . . . . . . . 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 1135 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣:(1...(♯‘𝑤))–1-1-onto𝑤 ∧ ∀𝑥 ∈ (1...(♯‘𝑤))∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 ↔ (𝑣𝑥) < (𝑣𝑦))))
3938simpld 495 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣:(1...(♯‘𝑤))–1-1-onto𝑤)
4016simprd 496 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤𝐵) → (♯‘𝑤) = 𝐾)
41 oveq2 7365 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑤) = 𝐾 → (1...(♯‘𝑤)) = (1...𝐾))
4241f1oeq2d 6780 . . . . . . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . . . . . 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 6784 . . . . . . . . . . . . . . 15 (𝑣:(1...𝐾)–1-1-onto𝑤𝑣:(1...𝐾)⟶𝑤)
4846, 47syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣:(1...𝐾)⟶𝑤)
4948ffnd 6669 . . . . . . . . . . . . 13 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣 Fn (1...𝐾))
50 ovexd 7392 . . . . . . . . . . . . 13 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (1...𝐾) ∈ V)
5149, 50fnexd 7168 . . . . . . . . . . . 12 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣 ∈ V)
52183adant3 1132 . . . . . . . . . . . . . 14 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑤 ⊆ (1...𝑁))
53 fss 6685 . . . . . . . . . . . . . 14 ((𝑣:(1...𝐾)⟶𝑤𝑤 ⊆ (1...𝑁)) → 𝑣:(1...𝐾)⟶(1...𝑁))
5448, 52, 53syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣:(1...𝐾)⟶(1...𝑁))
5538simprd 496 . . . . . . . . . . . . . . 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 3164 . . . . . . . . . . . . . . . 16 (((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) ∧ 𝑥 ∈ (1...(♯‘𝑤))) → (∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 ↔ (𝑣𝑥) < (𝑣𝑦)) → ∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦))))
5958ralimdva 3164 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤𝐵) ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (♯‘𝑤) = 𝐾)
62613impa 1110 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (♯‘𝑤) = 𝐾)
6362oveq2d 7373 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (1...(♯‘𝑤)) = (1...𝐾))
6463raleqdv 3313 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦)) ↔ ∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦))))
6564adantr 481 . . . . . . . . . . . . . . 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 3321 . . . . . . . . . . . . . 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 512 . . . . . . . . . . . 12 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦))))
69 feq1 6649 . . . . . . . . . . . . 13 (𝑓 = 𝑣 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑣:(1...𝐾)⟶(1...𝑁)))
70 fveq1 6841 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑣 → (𝑓𝑥) = (𝑣𝑥))
71 fveq1 6841 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑣 → (𝑓𝑦) = (𝑣𝑦))
7270, 71breq12d 5118 . . . . . . . . . . . . . . 15 (𝑓 = 𝑣 → ((𝑓𝑥) < (𝑓𝑦) ↔ (𝑣𝑥) < (𝑣𝑦)))
7372imbi2d 340 . . . . . . . . . . . . . 14 (𝑓 = 𝑣 → ((𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)) ↔ (𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦))))
74732ralbidv 3212 . . . . . . . . . . . . 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 631 . . . . . . . . . . . 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 3633 . . . . . . . . . . 11 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣 ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))})
774eleq2i 2829 . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . 17 (((𝜑𝑣𝐴) ∧ 𝑧 = 𝑣) → 𝑧 = 𝑣)
8180rneqd 5893 . . . . . . . . . . . . . . . 16 (((𝜑𝑣𝐴) ∧ 𝑧 = 𝑣) → ran 𝑧 = ran 𝑣)
82 simpr 485 . . . . . . . . . . . . . . . 16 ((𝜑𝑣𝐴) → 𝑣𝐴)
83 rnexg 7841 . . . . . . . . . . . . . . . . 17 (𝑣𝐴 → ran 𝑣 ∈ V)
8482, 83syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑣𝐴) → ran 𝑣 ∈ V)
8579, 81, 82, 84fvmptd 6955 . . . . . . . . . . . . . . 15 ((𝜑𝑣𝐴) → (𝐹𝑣) = ran 𝑣)
8685ex 413 . . . . . . . . . . . . . 14 (𝜑 → (𝑣𝐴 → (𝐹𝑣) = ran 𝑣))
87863ad2ant1 1133 . . . . . . . . . . . . 13 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣𝐴 → (𝐹𝑣) = ran 𝑣))
8878, 87mpd 15 . . . . . . . . . . . 12 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝐹𝑣) = ran 𝑣)
89 dff1o2 6789 . . . . . . . . . . . . . . 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 1144 . . . . . . . . . . . . 13 (𝑣:(1...𝐾)–1-1-onto𝑤 → ran 𝑣 = 𝑤)
9246, 91syl 17 . . . . . . . . . . . 12 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → ran 𝑣 = 𝑤)
9388, 92eqtrd 2776 . . . . . . . . . . 11 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝐹𝑣) = 𝑤)
9493eqcomd 2742 . . . . . . . . . 10 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑤 = (𝐹𝑣))
9578, 94jca 512 . . . . . . . . 9 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣𝐴𝑤 = (𝐹𝑣)))
96953expa 1118 . . . . . . . 8 (((𝜑𝑤𝐵) ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣𝐴𝑤 = (𝐹𝑣)))
9796ex 413 . . . . . . 7 ((𝜑𝑤𝐵) → (𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤) → (𝑣𝐴𝑤 = (𝐹𝑣))))
9897eximdv 1920 . . . . . 6 ((𝜑𝑤𝐵) → (∃𝑣 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤) → ∃𝑣(𝑣𝐴𝑤 = (𝐹𝑣))))
9935, 98mpd 15 . . . . 5 ((𝜑𝑤𝐵) → ∃𝑣(𝑣𝐴𝑤 = (𝐹𝑣)))
100 df-rex 3074 . . . . 5 (∃𝑣𝐴 𝑤 = (𝐹𝑣) ↔ ∃𝑣(𝑣𝐴𝑤 = (𝐹𝑣)))
10199, 100sylibr 233 . . . 4 ((𝜑𝑤𝐵) → ∃𝑣𝐴 𝑤 = (𝐹𝑣))
102101ralrimiva 3143 . . 3 (𝜑 → ∀𝑤𝐵𝑣𝐴 𝑤 = (𝐹𝑣))
10310, 102jca 512 . 2 (𝜑 → (𝐹:𝐴𝐵 ∧ ∀𝑤𝐵𝑣𝐴 𝑤 = (𝐹𝑣)))
104 dffo3 7052 . . 3 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑤𝐵𝑣𝐴 𝑤 = (𝐹𝑣)))
105104a1i 11 . 2 (𝜑 → (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑤𝐵𝑣𝐴 𝑤 = (𝐹𝑣))))
106103, 105mpbird 256 1 (𝜑𝐹:𝐴onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  wss 3910  𝒫 cpw 4560   class class class wbr 5105  cmpt 5188   Or wor 5544  ccnv 5632  ran crn 5634  Fun wfun 6490   Fn wfn 6491  wf 6492  1-1wf1 6493  ontowfo 6494  1-1-ontowf1o 6495  cfv 6496   Isom wiso 6497  (class class class)co 7357  Fincfn 8883  cr 11050  1c1 11052   < clt 11189  cn 12153  0cn0 12413  ...cfz 13424  chash 14230
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-hash 14231
This theorem is referenced by:  sticksstones4  40557
  Copyright terms: Public domain W3C validator