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Theorem sticksstones3 42130
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 42129 . . . 4 (𝜑𝐹:𝐴1-1𝐵)
7 df-f1 6568 . . . . . 6 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
87biimpi 216 . . . . 5 (𝐹:𝐴1-1𝐵 → (𝐹:𝐴𝐵 ∧ Fun 𝐹))
98simpld 494 . . . 4 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
106, 9syl 17 . . 3 (𝜑𝐹:𝐴𝐵)
113eleq2i 2831 . . . . . . . . . . . . . . . . . . . . 21 (𝑤𝐵𝑤 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾})
1211biimpi 216 . . . . . . . . . . . . . . . . . . . 20 (𝑤𝐵𝑤 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾})
1312adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤𝐵) → 𝑤 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾})
14 fveqeq2 6916 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑤 → ((♯‘𝑎) = 𝐾 ↔ (♯‘𝑤) = 𝐾))
1514elrab 3695 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾} ↔ (𝑤 ∈ 𝒫 (1...𝑁) ∧ (♯‘𝑤) = 𝐾))
1613, 15sylib 218 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤𝐵) → (𝑤 ∈ 𝒫 (1...𝑁) ∧ (♯‘𝑤) = 𝐾))
1716simpld 494 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝐵) → 𝑤 ∈ 𝒫 (1...𝑁))
1817elpwid 4614 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝐵) → 𝑤 ⊆ (1...𝑁))
1918sseld 3994 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝐵) → (𝑐𝑤𝑐 ∈ (1...𝑁)))
2019imp 406 . . . . . . . . . . . . . 14 (((𝜑𝑤𝐵) ∧ 𝑐𝑤) → 𝑐 ∈ (1...𝑁))
21203impa 1109 . . . . . . . . . . . . 13 ((𝜑𝑤𝐵𝑐𝑤) → 𝑐 ∈ (1...𝑁))
22 elfznn 13590 . . . . . . . . . . . . 13 (𝑐 ∈ (1...𝑁) → 𝑐 ∈ ℕ)
2321, 22syl 17 . . . . . . . . . . . 12 ((𝜑𝑤𝐵𝑐𝑤) → 𝑐 ∈ ℕ)
2423nnred 12279 . . . . . . . . . . 11 ((𝜑𝑤𝐵𝑐𝑤) → 𝑐 ∈ ℝ)
25243expa 1117 . . . . . . . . . 10 (((𝜑𝑤𝐵) ∧ 𝑐𝑤) → 𝑐 ∈ ℝ)
2625ex 412 . . . . . . . . 9 ((𝜑𝑤𝐵) → (𝑐𝑤𝑐 ∈ ℝ))
2726ssrdv 4001 . . . . . . . 8 ((𝜑𝑤𝐵) → 𝑤 ⊆ ℝ)
28 ltso 11339 . . . . . . . . 9 < Or ℝ
29 soss 5617 . . . . . . . . 9 (𝑤 ⊆ ℝ → ( < Or ℝ → < Or 𝑤))
3028, 29mpi 20 . . . . . . . 8 (𝑤 ⊆ ℝ → < Or 𝑤)
3127, 30syl 17 . . . . . . 7 ((𝜑𝑤𝐵) → < Or 𝑤)
32 fzfid 14011 . . . . . . . 8 ((𝜑𝑤𝐵) → (1...𝑁) ∈ Fin)
3332, 18ssfid 9299 . . . . . . 7 ((𝜑𝑤𝐵) → 𝑤 ∈ Fin)
34 fz1iso 14498 . . . . . . 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 6572 . . . . . . . . . . . . . . . . . . 19 (𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤) ↔ (𝑣:(1...(♯‘𝑤))–1-1-onto𝑤 ∧ ∀𝑥 ∈ (1...(♯‘𝑤))∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 ↔ (𝑣𝑥) < (𝑣𝑦))))
3736biimpi 216 . . . . . . . . . . . . . . . . . 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 7439 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑤) = 𝐾 → (1...(♯‘𝑤)) = (1...𝐾))
4241f1oeq2d 6845 . . . . . . . . . . . . . . . . . . 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 229 . . . . . . . . . . . . . . . . 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 6849 . . . . . . . . . . . . . . 15 (𝑣:(1...𝐾)–1-1-onto𝑤𝑣:(1...𝐾)⟶𝑤)
4846, 47syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣:(1...𝐾)⟶𝑤)
4948ffnd 6738 . . . . . . . . . . . . 13 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣 Fn (1...𝐾))
50 ovexd 7466 . . . . . . . . . . . . 13 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (1...𝐾) ∈ V)
5149, 50fnexd 7238 . . . . . . . . . . . 12 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣 ∈ V)
52183adant3 1131 . . . . . . . . . . . . . 14 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑤 ⊆ (1...𝑁))
53 fss 6753 . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → ∀𝑥 ∈ (1...(♯‘𝑤))∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 ↔ (𝑣𝑥) < (𝑣𝑦)))
56 biimp 215 . . . . . . . . . . . . . . . . . 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 3165 . . . . . . . . . . . . . . . 16 (((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) ∧ 𝑥 ∈ (1...(♯‘𝑤))) → (∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 ↔ (𝑣𝑥) < (𝑣𝑦)) → ∀𝑦 ∈ (1...(♯‘𝑤))(𝑥 < 𝑦 → (𝑣𝑥) < (𝑣𝑦))))
5958ralimdva 3165 . . . . . . . . . . . . . . 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 7447 . . . . . . . . . . . . . . 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 3330 . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . 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 6717 . . . . . . . . . . . . 13 (𝑓 = 𝑣 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑣:(1...𝐾)⟶(1...𝑁)))
70 fveq1 6906 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑣 → (𝑓𝑥) = (𝑣𝑥))
71 fveq1 6906 . . . . . . . . . . . . . . . 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 3219 . . . . . . . . . . . . 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 3684 . . . . . . . . . . 11 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → 𝑣 ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))})
774eleq2i 2831 . . . . . . . . . . 11 (𝑣𝐴𝑣 ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))})
7876, 77sylibr 234 . . . . . . . . . 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 5952 . . . . . . . . . . . . . . . 16 (((𝜑𝑣𝐴) ∧ 𝑧 = 𝑣) → ran 𝑧 = ran 𝑣)
82 simpr 484 . . . . . . . . . . . . . . . 16 ((𝜑𝑣𝐴) → 𝑣𝐴)
83 rnexg 7925 . . . . . . . . . . . . . . . . 17 (𝑣𝐴 → ran 𝑣 ∈ V)
8482, 83syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑣𝐴) → ran 𝑣 ∈ V)
8579, 81, 82, 84fvmptd 7023 . . . . . . . . . . . . . . 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 6854 . . . . . . . . . . . . . . 15 (𝑣:(1...𝐾)–1-1-onto𝑤 ↔ (𝑣 Fn (1...𝐾) ∧ Fun 𝑣 ∧ ran 𝑣 = 𝑤))
9089biimpi 216 . . . . . . . . . . . . . 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 2775 . . . . . . . . . . 11 ((𝜑𝑤𝐵𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤)) → (𝐹𝑣) = 𝑤)
9493eqcomd 2741 . . . . . . . . . 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 1915 . . . . . 6 ((𝜑𝑤𝐵) → (∃𝑣 𝑣 Isom < , < ((1...(♯‘𝑤)), 𝑤) → ∃𝑣(𝑣𝐴𝑤 = (𝐹𝑣))))
9935, 98mpd 15 . . . . 5 ((𝜑𝑤𝐵) → ∃𝑣(𝑣𝐴𝑤 = (𝐹𝑣)))
100 df-rex 3069 . . . . 5 (∃𝑣𝐴 𝑤 = (𝐹𝑣) ↔ ∃𝑣(𝑣𝐴𝑤 = (𝐹𝑣)))
10199, 100sylibr 234 . . . 4 ((𝜑𝑤𝐵) → ∃𝑣𝐴 𝑤 = (𝐹𝑣))
102101ralrimiva 3144 . . 3 (𝜑 → ∀𝑤𝐵𝑣𝐴 𝑤 = (𝐹𝑣))
10310, 102jca 511 . 2 (𝜑 → (𝐹:𝐴𝐵 ∧ ∀𝑤𝐵𝑣𝐴 𝑤 = (𝐹𝑣)))
104 dffo3 7122 . . 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 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  wss 3963  𝒫 cpw 4605   class class class wbr 5148  cmpt 5231   Or wor 5596  ccnv 5688  ran crn 5690  Fun wfun 6557   Fn wfn 6558  wf 6559  1-1wf1 6560  ontowfo 6561  1-1-ontowf1o 6562  cfv 6563   Isom wiso 6564  (class class class)co 7431  Fincfn 8984  cr 11152  1c1 11154   < clt 11293  cn 12264  0cn0 12524  ...cfz 13544  chash 14366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-hash 14367
This theorem is referenced by:  sticksstones4  42131
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