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Theorem sticksstones3 40602
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 40601 . . . 4 (𝜑𝐹:𝐴1-1𝐵)
7 df-f1 6502 . . . . . 6 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
87biimpi 215 . . . . 5 (𝐹:𝐴1-1𝐵 → (𝐹:𝐴𝐵 ∧ Fun 𝐹))
98simpld 496 . . . 4 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
106, 9syl 17 . . 3 (𝜑𝐹:𝐴𝐵)
113eleq2i 2826 . . . . . . . . . . . . . . . . . . . . 21 (𝑤𝐵𝑤 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾})
1211biimpi 215 . . . . . . . . . . . . . . . . . . . 20 (𝑤𝐵𝑤 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾})
1312adantl 483 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤𝐵) → 𝑤 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾})
14 fveqeq2 6852 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑤 → ((♯‘𝑎) = 𝐾 ↔ (♯‘𝑤) = 𝐾))
1514elrab 3646 . . . . . . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝐵) → 𝑤 ∈ 𝒫 (1...𝑁))
1817elpwid 4570 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝐵) → 𝑤 ⊆ (1...𝑁))
1918sseld 3944 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝐵) → (𝑐𝑤𝑐 ∈ (1...𝑁)))
2019imp 408 . . . . . . . . . . . . . 14 (((𝜑𝑤𝐵) ∧ 𝑐𝑤) → 𝑐 ∈ (1...𝑁))
21203impa 1111 . . . . . . . . . . . . 13 ((𝜑𝑤𝐵𝑐𝑤) → 𝑐 ∈ (1...𝑁))
22 elfznn 13476 . . . . . . . . . . . . 13 (𝑐 ∈ (1...𝑁) → 𝑐 ∈ ℕ)
2321, 22syl 17 . . . . . . . . . . . 12 ((𝜑𝑤𝐵𝑐𝑤) → 𝑐 ∈ ℕ)
2423nnred 12173 . . . . . . . . . . 11 ((𝜑𝑤𝐵𝑐𝑤) → 𝑐 ∈ ℝ)
25243expa 1119 . . . . . . . . . 10 (((𝜑𝑤𝐵) ∧ 𝑐𝑤) → 𝑐 ∈ ℝ)
2625ex 414 . . . . . . . . 9 ((𝜑𝑤𝐵) → (𝑐𝑤𝑐 ∈ ℝ))
2726ssrdv 3951 . . . . . . . 8 ((𝜑𝑤𝐵) → 𝑤 ⊆ ℝ)
28 ltso 11240 . . . . . . . . 9 < Or ℝ
29 soss 5566 . . . . . . . . 9 (𝑤 ⊆ ℝ → ( < Or ℝ → < Or 𝑤))
3028, 29mpi 20 . . . . . . . 8 (𝑤 ⊆ ℝ → < Or 𝑤)
3127, 30syl 17 . . . . . . 7 ((𝜑𝑤𝐵) → < Or 𝑤)
32 fzfid 13884 . . . . . . . 8 ((𝜑𝑤𝐵) → (1...𝑁) ∈ Fin)
3332, 18ssfid 9214 . . . . . . 7 ((𝜑𝑤𝐵) → 𝑤 ∈ Fin)
34 fz1iso 14367 . . . . . . 7 (( < Or 𝑤𝑤 ∈ Fin) → ∃𝑣 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤))
3531, 33, 34syl2anc 585 . . . . . 6 ((𝜑𝑤𝐵) → ∃𝑣 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤))
36 df-isom 6506 . . . . . . . . . . . . . . . . . . 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 1136 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣:(1...(♯‘𝑤))–1-1-onto𝑤 ∧ ∀𝑥 ∈ (1...(♯‘𝑤))∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 ↔ (𝑣𝑥) < (𝑣𝑦))))
3938simpld 496 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣:(1...(♯‘𝑤))–1-1-onto𝑤)
4016simprd 497 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤𝐵) → (♯‘𝑤) = 𝐾)
41 oveq2 7366 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑤) = 𝐾 → (1...(♯‘𝑤)) = (1...𝐾))
4241f1oeq2d 6781 . . . . . . . . . . . . . . . . . . 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 1133 . . . . . . . . . . . . . . . 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 6785 . . . . . . . . . . . . . . 15 (𝑣:(1...𝐾)–1-1-onto𝑤𝑣:(1...𝐾)⟶𝑤)
4846, 47syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣:(1...𝐾)⟶𝑤)
4948ffnd 6670 . . . . . . . . . . . . 13 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣 Fn (1...𝐾))
50 ovexd 7393 . . . . . . . . . . . . 13 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (1...𝐾) ∈ V)
5149, 50fnexd 7169 . . . . . . . . . . . 12 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣 ∈ V)
52183adant3 1133 . . . . . . . . . . . . . 14 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑤 ⊆ (1...𝑁))
53 fss 6686 . . . . . . . . . . . . . 14 ((𝑣:(1...𝐾)⟶𝑤𝑤 ⊆ (1...𝑁)) → 𝑣:(1...𝐾)⟶(1...𝑁))
5448, 52, 53syl2anc 585 . . . . . . . . . . . . 13 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣:(1...𝐾)⟶(1...𝑁))
5538simprd 497 . . . . . . . . . . . . . . 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 3161 . . . . . . . . . . . . . . . 16 (((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) ∧ 𝑥 ∈ (1...(♯‘𝑤))) → (∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 ↔ (𝑣𝑥) < (𝑣𝑦)) → ∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦))))
5958ralimdva 3161 . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤𝐵) ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (♯‘𝑤) = 𝐾)
62613impa 1111 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (♯‘𝑤) = 𝐾)
6362oveq2d 7374 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (1...(♯‘𝑤)) = (1...𝐾))
6463raleqdv 3312 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦)) ↔ ∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦))))
6564adantr 482 . . . . . . . . . . . . . . 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 3320 . . . . . . . . . . . . . 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 513 . . . . . . . . . . . 12 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦))))
69 feq1 6650 . . . . . . . . . . . . 13 (𝑓 = 𝑣 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑣:(1...𝐾)⟶(1...𝑁)))
70 fveq1 6842 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑣 → (𝑓𝑥) = (𝑣𝑥))
71 fveq1 6842 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑣 → (𝑓𝑦) = (𝑣𝑦))
7270, 71breq12d 5119 . . . . . . . . . . . . . . 15 (𝑓 = 𝑣 → ((𝑓𝑥) < (𝑓𝑦) ↔ (𝑣𝑥) < (𝑣𝑦)))
7372imbi2d 341 . . . . . . . . . . . . . 14 (𝑓 = 𝑣 → ((𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)) ↔ (𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦))))
74732ralbidv 3209 . . . . . . . . . . . . 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 632 . . . . . . . . . . . 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 3634 . . . . . . . . . . 11 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣 ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))})
774eleq2i 2826 . . . . . . . . . . 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 486 . . . . . . . . . . . . . . . . 17 (((𝜑𝑣𝐴) ∧ 𝑧 = 𝑣) → 𝑧 = 𝑣)
8180rneqd 5894 . . . . . . . . . . . . . . . 16 (((𝜑𝑣𝐴) ∧ 𝑧 = 𝑣) → ran 𝑧 = ran 𝑣)
82 simpr 486 . . . . . . . . . . . . . . . 16 ((𝜑𝑣𝐴) → 𝑣𝐴)
83 rnexg 7842 . . . . . . . . . . . . . . . . 17 (𝑣𝐴 → ran 𝑣 ∈ V)
8482, 83syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑣𝐴) → ran 𝑣 ∈ V)
8579, 81, 82, 84fvmptd 6956 . . . . . . . . . . . . . . 15 ((𝜑𝑣𝐴) → (𝐹𝑣) = ran 𝑣)
8685ex 414 . . . . . . . . . . . . . 14 (𝜑 → (𝑣𝐴 → (𝐹𝑣) = ran 𝑣))
87863ad2ant1 1134 . . . . . . . . . . . . 13 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣𝐴 → (𝐹𝑣) = ran 𝑣))
8878, 87mpd 15 . . . . . . . . . . . 12 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝐹𝑣) = ran 𝑣)
89 dff1o2 6790 . . . . . . . . . . . . . . 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 1145 . . . . . . . . . . . . 13 (𝑣:(1...𝐾)–1-1-onto𝑤 → ran 𝑣 = 𝑤)
9246, 91syl 17 . . . . . . . . . . . 12 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → ran 𝑣 = 𝑤)
9388, 92eqtrd 2773 . . . . . . . . . . 11 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝐹𝑣) = 𝑤)
9493eqcomd 2739 . . . . . . . . . 10 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑤 = (𝐹𝑣))
9578, 94jca 513 . . . . . . . . 9 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣𝐴𝑤 = (𝐹𝑣)))
96953expa 1119 . . . . . . . 8 (((𝜑𝑤𝐵) ∧ 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝑣𝐴𝑤 = (𝐹𝑣)))
9796ex 414 . . . . . . 7 ((𝜑𝑤𝐵) → (𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤) → (𝑣𝐴𝑤 = (𝐹𝑣))))
9897eximdv 1921 . . . . . 6 ((𝜑𝑤𝐵) → (∃𝑣 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤) → ∃𝑣(𝑣𝐴𝑤 = (𝐹𝑣))))
9935, 98mpd 15 . . . . 5 ((𝜑𝑤𝐵) → ∃𝑣(𝑣𝐴𝑤 = (𝐹𝑣)))
100 df-rex 3071 . . . . 5 (∃𝑣𝐴 𝑤 = (𝐹𝑣) ↔ ∃𝑣(𝑣𝐴𝑤 = (𝐹𝑣)))
10199, 100sylibr 233 . . . 4 ((𝜑𝑤𝐵) → ∃𝑣𝐴 𝑤 = (𝐹𝑣))
102101ralrimiva 3140 . . 3 (𝜑 → ∀𝑤𝐵𝑣𝐴 𝑤 = (𝐹𝑣))
10310, 102jca 513 . 2 (𝜑 → (𝐹:𝐴𝐵 ∧ ∀𝑤𝐵𝑣𝐴 𝑤 = (𝐹𝑣)))
104 dffo3 7053 . . 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 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wral 3061  wrex 3070  {crab 3406  Vcvv 3444  wss 3911  𝒫 cpw 4561   class class class wbr 5106  cmpt 5189   Or wor 5545  ccnv 5633  ran crn 5635  Fun wfun 6491   Fn wfn 6492  wf 6493  1-1wf1 6494  ontowfo 6495  1-1-ontowf1o 6496  cfv 6497   Isom wiso 6498  (class class class)co 7358  Fincfn 8886  cr 11055  1c1 11057   < clt 11194  cn 12158  0cn0 12418  ...cfz 13430  chash 14236
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9383  df-inf 9384  df-oi 9451  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-n0 12419  df-z 12505  df-uz 12769  df-fz 13431  df-hash 14237
This theorem is referenced by:  sticksstones4  40603
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