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Theorem f1o2ndf1 5772
 Description: The 2nd (second member of an ordered pair) function restricted to a one-to-one function 𝐹 is a one-to-one function of 𝐹 onto the range of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
Assertion
Ref Expression
f1o2ndf1 (𝐹:A1-1B → (2nd𝐹):𝐹1-1-onto→ran 𝐹)

Proof of Theorem f1o2ndf1
Dummy variables 𝑎 𝑏 v w x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1f 5017 . . 3 (𝐹:A1-1B𝐹:AB)
2 fo2ndf 5771 . . 3 (𝐹:AB → (2nd𝐹):𝐹onto→ran 𝐹)
31, 2syl 14 . 2 (𝐹:A1-1B → (2nd𝐹):𝐹onto→ran 𝐹)
4 f2ndf 5770 . . . . 5 (𝐹:AB → (2nd𝐹):𝐹B)
51, 4syl 14 . . . 4 (𝐹:A1-1B → (2nd𝐹):𝐹B)
6 fssxp 4983 . . . . . . 7 (𝐹:AB𝐹 ⊆ (A × B))
71, 6syl 14 . . . . . 6 (𝐹:A1-1B𝐹 ⊆ (A × B))
8 ssel2 2917 . . . . . . . . . . 11 ((𝐹 ⊆ (A × B) x 𝐹) → x (A × B))
9 elxp2 4290 . . . . . . . . . . 11 (x (A × B) ↔ 𝑎 A v B x = ⟨𝑎, v⟩)
108, 9sylib 127 . . . . . . . . . 10 ((𝐹 ⊆ (A × B) x 𝐹) → 𝑎 A v B x = ⟨𝑎, v⟩)
11 ssel2 2917 . . . . . . . . . . 11 ((𝐹 ⊆ (A × B) y 𝐹) → y (A × B))
12 elxp2 4290 . . . . . . . . . . 11 (y (A × B) ↔ 𝑏 A w B y = ⟨𝑏, w⟩)
1311, 12sylib 127 . . . . . . . . . 10 ((𝐹 ⊆ (A × B) y 𝐹) → 𝑏 A w B y = ⟨𝑏, w⟩)
1410, 13anim12dan 519 . . . . . . . . 9 ((𝐹 ⊆ (A × B) (x 𝐹 y 𝐹)) → (𝑎 A v B x = ⟨𝑎, v 𝑏 A w B y = ⟨𝑏, w⟩))
15 fvres 5123 . . . . . . . . . . . . . . . . . . . . . . . . 25 (⟨𝑎, v 𝐹 → ((2nd𝐹)‘⟨𝑎, v⟩) = (2nd ‘⟨𝑎, v⟩))
1615adantr 261 . . . . . . . . . . . . . . . . . . . . . . . 24 ((⟨𝑎, v 𝐹 𝑏, w 𝐹) → ((2nd𝐹)‘⟨𝑎, v⟩) = (2nd ‘⟨𝑎, v⟩))
1716adantr 261 . . . . . . . . . . . . . . . . . . . . . . 23 (((⟨𝑎, v 𝐹 𝑏, w 𝐹) ((𝑎 A v B) (𝑏 A w B))) → ((2nd𝐹)‘⟨𝑎, v⟩) = (2nd ‘⟨𝑎, v⟩))
18 fvres 5123 . . . . . . . . . . . . . . . . . . . . . . . 24 (⟨𝑏, w 𝐹 → ((2nd𝐹)‘⟨𝑏, w⟩) = (2nd ‘⟨𝑏, w⟩))
1918ad2antlr 462 . . . . . . . . . . . . . . . . . . . . . . 23 (((⟨𝑎, v 𝐹 𝑏, w 𝐹) ((𝑎 A v B) (𝑏 A w B))) → ((2nd𝐹)‘⟨𝑏, w⟩) = (2nd ‘⟨𝑏, w⟩))
2017, 19eqeq12d 2036 . . . . . . . . . . . . . . . . . . . . . 22 (((⟨𝑎, v 𝐹 𝑏, w 𝐹) ((𝑎 A v B) (𝑏 A w B))) → (((2nd𝐹)‘⟨𝑎, v⟩) = ((2nd𝐹)‘⟨𝑏, w⟩) ↔ (2nd ‘⟨𝑎, v⟩) = (2nd ‘⟨𝑏, w⟩)))
21 vex 2538 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑎 V
22 vex 2538 . . . . . . . . . . . . . . . . . . . . . . . . 25 v V
2321, 22op2nd 5697 . . . . . . . . . . . . . . . . . . . . . . . 24 (2nd ‘⟨𝑎, v⟩) = v
24 vex 2538 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑏 V
25 vex 2538 . . . . . . . . . . . . . . . . . . . . . . . . 25 w V
2624, 25op2nd 5697 . . . . . . . . . . . . . . . . . . . . . . . 24 (2nd ‘⟨𝑏, w⟩) = w
2723, 26eqeq12i 2035 . . . . . . . . . . . . . . . . . . . . . . 23 ((2nd ‘⟨𝑎, v⟩) = (2nd ‘⟨𝑏, w⟩) ↔ v = w)
28 f1fun 5019 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐹:A1-1B → Fun 𝐹)
29 funopfv 5138 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Fun 𝐹 → (⟨𝑎, v 𝐹 → (𝐹𝑎) = v))
30 funopfv 5138 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Fun 𝐹 → (⟨𝑏, w 𝐹 → (𝐹𝑏) = w))
3129, 30anim12d 318 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Fun 𝐹 → ((⟨𝑎, v 𝐹 𝑏, w 𝐹) → ((𝐹𝑎) = v (𝐹𝑏) = w)))
3228, 31syl 14 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐹:A1-1B → ((⟨𝑎, v 𝐹 𝑏, w 𝐹) → ((𝐹𝑎) = v (𝐹𝑏) = w)))
33 eqcom 2024 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝐹𝑎) = vv = (𝐹𝑎))
3433biimpi 113 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐹𝑎) = vv = (𝐹𝑎))
35 eqcom 2024 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝐹𝑏) = ww = (𝐹𝑏))
3635biimpi 113 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐹𝑏) = ww = (𝐹𝑏))
3734, 36eqeqan12d 2037 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝐹𝑎) = v (𝐹𝑏) = w) → (v = w ↔ (𝐹𝑎) = (𝐹𝑏)))
38 simpl 102 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎 A v B) → 𝑎 A)
39 simpl 102 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑏 A w B) → 𝑏 A)
4038, 39anim12i 321 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((𝑎 A v B) (𝑏 A w B)) → (𝑎 A 𝑏 A))
41 f1veqaeq 5333 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝐹:A1-1B (𝑎 A 𝑏 A)) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
4240, 41sylan2 270 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝐹:A1-1B ((𝑎 A v B) (𝑏 A w B))) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
43 opeq12 3525 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑎 = 𝑏 v = w) → ⟨𝑎, v⟩ = ⟨𝑏, w⟩)
4443ex 108 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑎 = 𝑏 → (v = w → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))
4542, 44syl6 29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝐹:A1-1B ((𝑎 A v B) (𝑏 A w B))) → ((𝐹𝑎) = (𝐹𝑏) → (v = w → ⟨𝑎, v⟩ = ⟨𝑏, w⟩)))
4645com23 72 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝐹:A1-1B ((𝑎 A v B) (𝑏 A w B))) → (v = w → ((𝐹𝑎) = (𝐹𝑏) → ⟨𝑎, v⟩ = ⟨𝑏, w⟩)))
4746ex 108 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝐹:A1-1B → (((𝑎 A v B) (𝑏 A w B)) → (v = w → ((𝐹𝑎) = (𝐹𝑏) → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
4847com14 82 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐹𝑎) = (𝐹𝑏) → (((𝑎 A v B) (𝑏 A w B)) → (v = w → (𝐹:A1-1B → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
4937, 48syl6bi 152 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝐹𝑎) = v (𝐹𝑏) = w) → (v = w → (((𝑎 A v B) (𝑏 A w B)) → (v = w → (𝐹:A1-1B → ⟨𝑎, v⟩ = ⟨𝑏, w⟩)))))
5049com14 82 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (v = w → (v = w → (((𝑎 A v B) (𝑏 A w B)) → (((𝐹𝑎) = v (𝐹𝑏) = w) → (𝐹:A1-1B → ⟨𝑎, v⟩ = ⟨𝑏, w⟩)))))
5150pm2.43i 43 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (v = w → (((𝑎 A v B) (𝑏 A w B)) → (((𝐹𝑎) = v (𝐹𝑏) = w) → (𝐹:A1-1B → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
5251com14 82 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐹:A1-1B → (((𝑎 A v B) (𝑏 A w B)) → (((𝐹𝑎) = v (𝐹𝑏) = w) → (v = w → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
5352com23 72 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐹:A1-1B → (((𝐹𝑎) = v (𝐹𝑏) = w) → (((𝑎 A v B) (𝑏 A w B)) → (v = w → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
5432, 53syld 40 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐹:A1-1B → ((⟨𝑎, v 𝐹 𝑏, w 𝐹) → (((𝑎 A v B) (𝑏 A w B)) → (v = w → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
5554com13 74 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑎 A v B) (𝑏 A w B)) → ((⟨𝑎, v 𝐹 𝑏, w 𝐹) → (𝐹:A1-1B → (v = w → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
5655impcom 116 . . . . . . . . . . . . . . . . . . . . . . . 24 (((⟨𝑎, v 𝐹 𝑏, w 𝐹) ((𝑎 A v B) (𝑏 A w B))) → (𝐹:A1-1B → (v = w → ⟨𝑎, v⟩ = ⟨𝑏, w⟩)))
5756com23 72 . . . . . . . . . . . . . . . . . . . . . . 23 (((⟨𝑎, v 𝐹 𝑏, w 𝐹) ((𝑎 A v B) (𝑏 A w B))) → (v = w → (𝐹:A1-1B → ⟨𝑎, v⟩ = ⟨𝑏, w⟩)))
5827, 57syl5bi 141 . . . . . . . . . . . . . . . . . . . . . 22 (((⟨𝑎, v 𝐹 𝑏, w 𝐹) ((𝑎 A v B) (𝑏 A w B))) → ((2nd ‘⟨𝑎, v⟩) = (2nd ‘⟨𝑏, w⟩) → (𝐹:A1-1B → ⟨𝑎, v⟩ = ⟨𝑏, w⟩)))
5920, 58sylbid 139 . . . . . . . . . . . . . . . . . . . . 21 (((⟨𝑎, v 𝐹 𝑏, w 𝐹) ((𝑎 A v B) (𝑏 A w B))) → (((2nd𝐹)‘⟨𝑎, v⟩) = ((2nd𝐹)‘⟨𝑏, w⟩) → (𝐹:A1-1B → ⟨𝑎, v⟩ = ⟨𝑏, w⟩)))
6059com23 72 . . . . . . . . . . . . . . . . . . . 20 (((⟨𝑎, v 𝐹 𝑏, w 𝐹) ((𝑎 A v B) (𝑏 A w B))) → (𝐹:A1-1B → (((2nd𝐹)‘⟨𝑎, v⟩) = ((2nd𝐹)‘⟨𝑏, w⟩) → ⟨𝑎, v⟩ = ⟨𝑏, w⟩)))
6160ex 108 . . . . . . . . . . . . . . . . . . 19 ((⟨𝑎, v 𝐹 𝑏, w 𝐹) → (((𝑎 A v B) (𝑏 A w B)) → (𝐹:A1-1B → (((2nd𝐹)‘⟨𝑎, v⟩) = ((2nd𝐹)‘⟨𝑏, w⟩) → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
6261adantl 262 . . . . . . . . . . . . . . . . . 18 ((𝐹 ⊆ (A × B) (⟨𝑎, v 𝐹 𝑏, w 𝐹)) → (((𝑎 A v B) (𝑏 A w B)) → (𝐹:A1-1B → (((2nd𝐹)‘⟨𝑎, v⟩) = ((2nd𝐹)‘⟨𝑏, w⟩) → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
6362com12 27 . . . . . . . . . . . . . . . . 17 (((𝑎 A v B) (𝑏 A w B)) → ((𝐹 ⊆ (A × B) (⟨𝑎, v 𝐹 𝑏, w 𝐹)) → (𝐹:A1-1B → (((2nd𝐹)‘⟨𝑎, v⟩) = ((2nd𝐹)‘⟨𝑏, w⟩) → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
6463adantlr 449 . . . . . . . . . . . . . . . 16 ((((𝑎 A v B) x = ⟨𝑎, v⟩) (𝑏 A w B)) → ((𝐹 ⊆ (A × B) (⟨𝑎, v 𝐹 𝑏, w 𝐹)) → (𝐹:A1-1B → (((2nd𝐹)‘⟨𝑎, v⟩) = ((2nd𝐹)‘⟨𝑏, w⟩) → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
6564adantr 261 . . . . . . . . . . . . . . 15 (((((𝑎 A v B) x = ⟨𝑎, v⟩) (𝑏 A w B)) y = ⟨𝑏, w⟩) → ((𝐹 ⊆ (A × B) (⟨𝑎, v 𝐹 𝑏, w 𝐹)) → (𝐹:A1-1B → (((2nd𝐹)‘⟨𝑎, v⟩) = ((2nd𝐹)‘⟨𝑏, w⟩) → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
66 eleq1 2082 . . . . . . . . . . . . . . . . . 18 (x = ⟨𝑎, v⟩ → (x 𝐹 ↔ ⟨𝑎, v 𝐹))
6766ad2antlr 462 . . . . . . . . . . . . . . . . 17 ((((𝑎 A v B) x = ⟨𝑎, v⟩) (𝑏 A w B)) → (x 𝐹 ↔ ⟨𝑎, v 𝐹))
68 eleq1 2082 . . . . . . . . . . . . . . . . 17 (y = ⟨𝑏, w⟩ → (y 𝐹 ↔ ⟨𝑏, w 𝐹))
6967, 68bi2anan9 526 . . . . . . . . . . . . . . . 16 (((((𝑎 A v B) x = ⟨𝑎, v⟩) (𝑏 A w B)) y = ⟨𝑏, w⟩) → ((x 𝐹 y 𝐹) ↔ (⟨𝑎, v 𝐹 𝑏, w 𝐹)))
7069anbi2d 440 . . . . . . . . . . . . . . 15 (((((𝑎 A v B) x = ⟨𝑎, v⟩) (𝑏 A w B)) y = ⟨𝑏, w⟩) → ((𝐹 ⊆ (A × B) (x 𝐹 y 𝐹)) ↔ (𝐹 ⊆ (A × B) (⟨𝑎, v 𝐹 𝑏, w 𝐹))))
71 fveq2 5103 . . . . . . . . . . . . . . . . . . 19 (x = ⟨𝑎, v⟩ → ((2nd𝐹)‘x) = ((2nd𝐹)‘⟨𝑎, v⟩))
7271ad2antlr 462 . . . . . . . . . . . . . . . . . 18 ((((𝑎 A v B) x = ⟨𝑎, v⟩) (𝑏 A w B)) → ((2nd𝐹)‘x) = ((2nd𝐹)‘⟨𝑎, v⟩))
73 fveq2 5103 . . . . . . . . . . . . . . . . . 18 (y = ⟨𝑏, w⟩ → ((2nd𝐹)‘y) = ((2nd𝐹)‘⟨𝑏, w⟩))
7472, 73eqeqan12d 2037 . . . . . . . . . . . . . . . . 17 (((((𝑎 A v B) x = ⟨𝑎, v⟩) (𝑏 A w B)) y = ⟨𝑏, w⟩) → (((2nd𝐹)‘x) = ((2nd𝐹)‘y) ↔ ((2nd𝐹)‘⟨𝑎, v⟩) = ((2nd𝐹)‘⟨𝑏, w⟩)))
75 simpllr 474 . . . . . . . . . . . . . . . . . 18 (((((𝑎 A v B) x = ⟨𝑎, v⟩) (𝑏 A w B)) y = ⟨𝑏, w⟩) → x = ⟨𝑎, v⟩)
76 simpr 103 . . . . . . . . . . . . . . . . . 18 (((((𝑎 A v B) x = ⟨𝑎, v⟩) (𝑏 A w B)) y = ⟨𝑏, w⟩) → y = ⟨𝑏, w⟩)
7775, 76eqeq12d 2036 . . . . . . . . . . . . . . . . 17 (((((𝑎 A v B) x = ⟨𝑎, v⟩) (𝑏 A w B)) y = ⟨𝑏, w⟩) → (x = y ↔ ⟨𝑎, v⟩ = ⟨𝑏, w⟩))
7874, 77imbi12d 223 . . . . . . . . . . . . . . . 16 (((((𝑎 A v B) x = ⟨𝑎, v⟩) (𝑏 A w B)) y = ⟨𝑏, w⟩) → ((((2nd𝐹)‘x) = ((2nd𝐹)‘y) → x = y) ↔ (((2nd𝐹)‘⟨𝑎, v⟩) = ((2nd𝐹)‘⟨𝑏, w⟩) → ⟨𝑎, v⟩ = ⟨𝑏, w⟩)))
7978imbi2d 219 . . . . . . . . . . . . . . 15 (((((𝑎 A v B) x = ⟨𝑎, v⟩) (𝑏 A w B)) y = ⟨𝑏, w⟩) → ((𝐹:A1-1B → (((2nd𝐹)‘x) = ((2nd𝐹)‘y) → x = y)) ↔ (𝐹:A1-1B → (((2nd𝐹)‘⟨𝑎, v⟩) = ((2nd𝐹)‘⟨𝑏, w⟩) → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
8065, 70, 793imtr4d 192 . . . . . . . . . . . . . 14 (((((𝑎 A v B) x = ⟨𝑎, v⟩) (𝑏 A w B)) y = ⟨𝑏, w⟩) → ((𝐹 ⊆ (A × B) (x 𝐹 y 𝐹)) → (𝐹:A1-1B → (((2nd𝐹)‘x) = ((2nd𝐹)‘y) → x = y))))
8180ex 108 . . . . . . . . . . . . 13 ((((𝑎 A v B) x = ⟨𝑎, v⟩) (𝑏 A w B)) → (y = ⟨𝑏, w⟩ → ((𝐹 ⊆ (A × B) (x 𝐹 y 𝐹)) → (𝐹:A1-1B → (((2nd𝐹)‘x) = ((2nd𝐹)‘y) → x = y)))))
8281rexlimdvva 2418 . . . . . . . . . . . 12 (((𝑎 A v B) x = ⟨𝑎, v⟩) → (𝑏 A w B y = ⟨𝑏, w⟩ → ((𝐹 ⊆ (A × B) (x 𝐹 y 𝐹)) → (𝐹:A1-1B → (((2nd𝐹)‘x) = ((2nd𝐹)‘y) → x = y)))))
8382ex 108 . . . . . . . . . . 11 ((𝑎 A v B) → (x = ⟨𝑎, v⟩ → (𝑏 A w B y = ⟨𝑏, w⟩ → ((𝐹 ⊆ (A × B) (x 𝐹 y 𝐹)) → (𝐹:A1-1B → (((2nd𝐹)‘x) = ((2nd𝐹)‘y) → x = y))))))
8483rexlimivv 2416 . . . . . . . . . 10 (𝑎 A v B x = ⟨𝑎, v⟩ → (𝑏 A w B y = ⟨𝑏, w⟩ → ((𝐹 ⊆ (A × B) (x 𝐹 y 𝐹)) → (𝐹:A1-1B → (((2nd𝐹)‘x) = ((2nd𝐹)‘y) → x = y)))))
8584imp 115 . . . . . . . . 9 ((𝑎 A v B x = ⟨𝑎, v 𝑏 A w B y = ⟨𝑏, w⟩) → ((𝐹 ⊆ (A × B) (x 𝐹 y 𝐹)) → (𝐹:A1-1B → (((2nd𝐹)‘x) = ((2nd𝐹)‘y) → x = y))))
8614, 85mpcom 32 . . . . . . . 8 ((𝐹 ⊆ (A × B) (x 𝐹 y 𝐹)) → (𝐹:A1-1B → (((2nd𝐹)‘x) = ((2nd𝐹)‘y) → x = y)))
8786ex 108 . . . . . . 7 (𝐹 ⊆ (A × B) → ((x 𝐹 y 𝐹) → (𝐹:A1-1B → (((2nd𝐹)‘x) = ((2nd𝐹)‘y) → x = y))))
8887com23 72 . . . . . 6 (𝐹 ⊆ (A × B) → (𝐹:A1-1B → ((x 𝐹 y 𝐹) → (((2nd𝐹)‘x) = ((2nd𝐹)‘y) → x = y))))
897, 88mpcom 32 . . . . 5 (𝐹:A1-1B → ((x 𝐹 y 𝐹) → (((2nd𝐹)‘x) = ((2nd𝐹)‘y) → x = y)))
9089ralrimivv 2378 . . . 4 (𝐹:A1-1Bx 𝐹 y 𝐹 (((2nd𝐹)‘x) = ((2nd𝐹)‘y) → x = y))
91 dff13 5332 . . . 4 ((2nd𝐹):𝐹1-1B ↔ ((2nd𝐹):𝐹B x 𝐹 y 𝐹 (((2nd𝐹)‘x) = ((2nd𝐹)‘y) → x = y)))
925, 90, 91sylanbrc 396 . . 3 (𝐹:A1-1B → (2nd𝐹):𝐹1-1B)
93 df-f1 4834 . . . 4 ((2nd𝐹):𝐹1-1B ↔ ((2nd𝐹):𝐹B Fun (2nd𝐹)))
9493simprbi 260 . . 3 ((2nd𝐹):𝐹1-1B → Fun (2nd𝐹))
9592, 94syl 14 . 2 (𝐹:A1-1B → Fun (2nd𝐹))
96 dff1o3 5057 . 2 ((2nd𝐹):𝐹1-1-onto→ran 𝐹 ↔ ((2nd𝐹):𝐹onto→ran 𝐹 Fun (2nd𝐹)))
973, 95, 96sylanbrc 396 1 (𝐹:A1-1B → (2nd𝐹):𝐹1-1-onto→ran 𝐹)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228   ∈ wcel 1374  ∀wral 2284  ∃wrex 2285   ⊆ wss 2894  ⟨cop 3353   × cxp 4270  ◡ccnv 4271  ran crn 4273   ↾ cres 4274  Fun wfun 4823  ⟶wf 4825  –1-1→wf1 4826  –onto→wfo 4827  –1-1-onto→wf1o 4828  ‘cfv 4829  2nd c2nd 5689 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-2nd 5691 This theorem is referenced by: (None)
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