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Theorem f1o2ndf1 8054
Description: The 2nd (second component of an ordered pair) function restricted to a one-to-one function 𝐹 is a one-to-one function from 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 6738 . . 3 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
2 fo2ndf 8053 . . 3 (𝐹:𝐴𝐵 → (2nd𝐹):𝐹onto→ran 𝐹)
31, 2syl 17 . 2 (𝐹:𝐴1-1𝐵 → (2nd𝐹):𝐹onto→ran 𝐹)
4 f2ndf 8052 . . . . 5 (𝐹:𝐴𝐵 → (2nd𝐹):𝐹𝐵)
51, 4syl 17 . . . 4 (𝐹:𝐴1-1𝐵 → (2nd𝐹):𝐹𝐵)
6 fssxp 6696 . . . . . . 7 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
71, 6syl 17 . . . . . 6 (𝐹:𝐴1-1𝐵𝐹 ⊆ (𝐴 × 𝐵))
8 ssel2 3939 . . . . . . . . . . 11 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ 𝑥𝐹) → 𝑥 ∈ (𝐴 × 𝐵))
9 elxp2 5657 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 × 𝐵) ↔ ∃𝑎𝐴𝑣𝐵 𝑥 = ⟨𝑎, 𝑣⟩)
108, 9sylib 217 . . . . . . . . . 10 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ 𝑥𝐹) → ∃𝑎𝐴𝑣𝐵 𝑥 = ⟨𝑎, 𝑣⟩)
11 ssel2 3939 . . . . . . . . . . 11 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ 𝑦𝐹) → 𝑦 ∈ (𝐴 × 𝐵))
12 elxp2 5657 . . . . . . . . . . 11 (𝑦 ∈ (𝐴 × 𝐵) ↔ ∃𝑏𝐴𝑤𝐵 𝑦 = ⟨𝑏, 𝑤⟩)
1311, 12sylib 217 . . . . . . . . . 10 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ 𝑦𝐹) → ∃𝑏𝐴𝑤𝐵 𝑦 = ⟨𝑏, 𝑤⟩)
1410, 13anim12dan 619 . . . . . . . . 9 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) → (∃𝑎𝐴𝑣𝐵 𝑥 = ⟨𝑎, 𝑣⟩ ∧ ∃𝑏𝐴𝑤𝐵 𝑦 = ⟨𝑏, 𝑤⟩))
15 fvres 6861 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨𝑎, 𝑣⟩ ∈ 𝐹 → ((2nd𝐹)‘⟨𝑎, 𝑣⟩) = (2nd ‘⟨𝑎, 𝑣⟩))
1615ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . 22 (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → ((2nd𝐹)‘⟨𝑎, 𝑣⟩) = (2nd ‘⟨𝑎, 𝑣⟩))
17 fvres 6861 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨𝑏, 𝑤⟩ ∈ 𝐹 → ((2nd𝐹)‘⟨𝑏, 𝑤⟩) = (2nd ‘⟨𝑏, 𝑤⟩))
1817ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . 22 (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → ((2nd𝐹)‘⟨𝑏, 𝑤⟩) = (2nd ‘⟨𝑏, 𝑤⟩))
1916, 18eqeq12d 2752 . . . . . . . . . . . . . . . . . . . . 21 (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → (((2nd𝐹)‘⟨𝑎, 𝑣⟩) = ((2nd𝐹)‘⟨𝑏, 𝑤⟩) ↔ (2nd ‘⟨𝑎, 𝑣⟩) = (2nd ‘⟨𝑏, 𝑤⟩)))
20 vex 3449 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑎 ∈ V
21 vex 3449 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑣 ∈ V
2220, 21op2nd 7930 . . . . . . . . . . . . . . . . . . . . . . 23 (2nd ‘⟨𝑎, 𝑣⟩) = 𝑣
23 vex 3449 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑏 ∈ V
24 vex 3449 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑤 ∈ V
2523, 24op2nd 7930 . . . . . . . . . . . . . . . . . . . . . . 23 (2nd ‘⟨𝑏, 𝑤⟩) = 𝑤
2622, 25eqeq12i 2754 . . . . . . . . . . . . . . . . . . . . . 22 ((2nd ‘⟨𝑎, 𝑣⟩) = (2nd ‘⟨𝑏, 𝑤⟩) ↔ 𝑣 = 𝑤)
27 f1fun 6740 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐹:𝐴1-1𝐵 → Fun 𝐹)
28 funopfv 6894 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Fun 𝐹 → (⟨𝑎, 𝑣⟩ ∈ 𝐹 → (𝐹𝑎) = 𝑣))
29 funopfv 6894 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Fun 𝐹 → (⟨𝑏, 𝑤⟩ ∈ 𝐹 → (𝐹𝑏) = 𝑤))
3028, 29anim12d 609 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Fun 𝐹 → ((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) → ((𝐹𝑎) = 𝑣 ∧ (𝐹𝑏) = 𝑤)))
3127, 30syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐹:𝐴1-1𝐵 → ((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) → ((𝐹𝑎) = 𝑣 ∧ (𝐹𝑏) = 𝑤)))
32 eqcom 2743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐹𝑎) = 𝑣𝑣 = (𝐹𝑎))
3332biimpi 215 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐹𝑎) = 𝑣𝑣 = (𝐹𝑎))
34 eqcom 2743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐹𝑏) = 𝑤𝑤 = (𝐹𝑏))
3534biimpi 215 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐹𝑏) = 𝑤𝑤 = (𝐹𝑏))
3633, 35eqeqan12d 2750 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝐹𝑎) = 𝑣 ∧ (𝐹𝑏) = 𝑤) → (𝑣 = 𝑤 ↔ (𝐹𝑎) = (𝐹𝑏)))
37 simpl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑎𝐴𝑣𝐵) → 𝑎𝐴)
38 simpl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑏𝐴𝑤𝐵) → 𝑏𝐴)
3937, 38anim12i 613 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → (𝑎𝐴𝑏𝐴))
40 f1veqaeq 7204 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝐹:𝐴1-1𝐵 ∧ (𝑎𝐴𝑏𝐴)) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
4139, 40sylan2 593 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝐹:𝐴1-1𝐵 ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
42 opeq12 4832 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑎 = 𝑏𝑣 = 𝑤) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)
4342ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑎 = 𝑏 → (𝑣 = 𝑤 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))
4441, 43syl6 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝐹:𝐴1-1𝐵 ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → ((𝐹𝑎) = (𝐹𝑏) → (𝑣 = 𝑤 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)))
4544com23 86 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐹:𝐴1-1𝐵 ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → (𝑣 = 𝑤 → ((𝐹𝑎) = (𝐹𝑏) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)))
4645ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝐹:𝐴1-1𝐵 → (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → (𝑣 = 𝑤 → ((𝐹𝑎) = (𝐹𝑏) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
4746com14 96 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝐹𝑎) = (𝐹𝑏) → (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → (𝑣 = 𝑤 → (𝐹:𝐴1-1𝐵 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
4836, 47syl6bi 252 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝐹𝑎) = 𝑣 ∧ (𝐹𝑏) = 𝑤) → (𝑣 = 𝑤 → (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → (𝑣 = 𝑤 → (𝐹:𝐴1-1𝐵 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)))))
4948com14 96 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑣 = 𝑤 → (𝑣 = 𝑤 → (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → (((𝐹𝑎) = 𝑣 ∧ (𝐹𝑏) = 𝑤) → (𝐹:𝐴1-1𝐵 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)))))
5049pm2.43i 52 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑣 = 𝑤 → (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → (((𝐹𝑎) = 𝑣 ∧ (𝐹𝑏) = 𝑤) → (𝐹:𝐴1-1𝐵 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
5150com14 96 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐹:𝐴1-1𝐵 → (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → (((𝐹𝑎) = 𝑣 ∧ (𝐹𝑏) = 𝑤) → (𝑣 = 𝑤 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
5251com23 86 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐹:𝐴1-1𝐵 → (((𝐹𝑎) = 𝑣 ∧ (𝐹𝑏) = 𝑤) → (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → (𝑣 = 𝑤 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
5331, 52syld 47 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐹:𝐴1-1𝐵 → ((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) → (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → (𝑣 = 𝑤 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
5453com13 88 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → ((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) → (𝐹:𝐴1-1𝐵 → (𝑣 = 𝑤 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
5554impcom 408 . . . . . . . . . . . . . . . . . . . . . . 23 (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → (𝐹:𝐴1-1𝐵 → (𝑣 = 𝑤 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)))
5655com23 86 . . . . . . . . . . . . . . . . . . . . . 22 (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → (𝑣 = 𝑤 → (𝐹:𝐴1-1𝐵 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)))
5726, 56biimtrid 241 . . . . . . . . . . . . . . . . . . . . 21 (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → ((2nd ‘⟨𝑎, 𝑣⟩) = (2nd ‘⟨𝑏, 𝑤⟩) → (𝐹:𝐴1-1𝐵 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)))
5819, 57sylbid 239 . . . . . . . . . . . . . . . . . . . 20 (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → (((2nd𝐹)‘⟨𝑎, 𝑣⟩) = ((2nd𝐹)‘⟨𝑏, 𝑤⟩) → (𝐹:𝐴1-1𝐵 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)))
5958com23 86 . . . . . . . . . . . . . . . . . . 19 (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘⟨𝑎, 𝑣⟩) = ((2nd𝐹)‘⟨𝑏, 𝑤⟩) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)))
6059ex 413 . . . . . . . . . . . . . . . . . 18 ((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) → (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘⟨𝑎, 𝑣⟩) = ((2nd𝐹)‘⟨𝑏, 𝑤⟩) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
6160adantl 482 . . . . . . . . . . . . . . . . 17 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹)) → (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘⟨𝑎, 𝑣⟩) = ((2nd𝐹)‘⟨𝑏, 𝑤⟩) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
6261com12 32 . . . . . . . . . . . . . . . 16 (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘⟨𝑎, 𝑣⟩) = ((2nd𝐹)‘⟨𝑏, 𝑤⟩) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
6362ad4ant13 749 . . . . . . . . . . . . . . 15 (((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) ∧ 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⟩))))
64 eleq1 2825 . . . . . . . . . . . . . . . . . 18 (𝑥 = ⟨𝑎, 𝑣⟩ → (𝑥𝐹 ↔ ⟨𝑎, 𝑣⟩ ∈ 𝐹))
6564ad2antlr 725 . . . . . . . . . . . . . . . . 17 ((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) → (𝑥𝐹 ↔ ⟨𝑎, 𝑣⟩ ∈ 𝐹))
66 eleq1 2825 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑏, 𝑤⟩ → (𝑦𝐹 ↔ ⟨𝑏, 𝑤⟩ ∈ 𝐹))
6765, 66bi2anan9 637 . . . . . . . . . . . . . . . 16 (((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → ((𝑥𝐹𝑦𝐹) ↔ (⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹)))
6867anbi2d 629 . . . . . . . . . . . . . . 15 (((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ (⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹))))
69 fveq2 6842 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ⟨𝑎, 𝑣⟩ → ((2nd𝐹)‘𝑥) = ((2nd𝐹)‘⟨𝑎, 𝑣⟩))
7069ad2antlr 725 . . . . . . . . . . . . . . . . . 18 ((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) → ((2nd𝐹)‘𝑥) = ((2nd𝐹)‘⟨𝑎, 𝑣⟩))
71 fveq2 6842 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨𝑏, 𝑤⟩ → ((2nd𝐹)‘𝑦) = ((2nd𝐹)‘⟨𝑏, 𝑤⟩))
7270, 71eqeqan12d 2750 . . . . . . . . . . . . . . . . 17 (((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) ↔ ((2nd𝐹)‘⟨𝑎, 𝑣⟩) = ((2nd𝐹)‘⟨𝑏, 𝑤⟩)))
73 simpllr 774 . . . . . . . . . . . . . . . . . 18 (((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → 𝑥 = ⟨𝑎, 𝑣⟩)
74 simpr 485 . . . . . . . . . . . . . . . . . 18 (((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → 𝑦 = ⟨𝑏, 𝑤⟩)
7573, 74eqeq12d 2752 . . . . . . . . . . . . . . . . 17 (((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → (𝑥 = 𝑦 ↔ ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))
7672, 75imbi12d 344 . . . . . . . . . . . . . . . 16 (((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → ((((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦) ↔ (((2nd𝐹)‘⟨𝑎, 𝑣⟩) = ((2nd𝐹)‘⟨𝑏, 𝑤⟩) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)))
7776imbi2d 340 . . . . . . . . . . . . . . 15 (((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) ∧ 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⟩))))
7863, 68, 773imtr4d 293 . . . . . . . . . . . . . 14 (((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦))))
7978ex 413 . . . . . . . . . . . . 13 ((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) → (𝑦 = ⟨𝑏, 𝑤⟩ → ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦)))))
8079rexlimdvva 3205 . . . . . . . . . . . 12 (((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) → (∃𝑏𝐴𝑤𝐵 𝑦 = ⟨𝑏, 𝑤⟩ → ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦)))))
8180ex 413 . . . . . . . . . . 11 ((𝑎𝐴𝑣𝐵) → (𝑥 = ⟨𝑎, 𝑣⟩ → (∃𝑏𝐴𝑤𝐵 𝑦 = ⟨𝑏, 𝑤⟩ → ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦))))))
8281rexlimivv 3196 . . . . . . . . . 10 (∃𝑎𝐴𝑣𝐵 𝑥 = ⟨𝑎, 𝑣⟩ → (∃𝑏𝐴𝑤𝐵 𝑦 = ⟨𝑏, 𝑤⟩ → ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦)))))
8382imp 407 . . . . . . . . 9 ((∃𝑎𝐴𝑣𝐵 𝑥 = ⟨𝑎, 𝑣⟩ ∧ ∃𝑏𝐴𝑤𝐵 𝑦 = ⟨𝑏, 𝑤⟩) → ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦))))
8414, 83mpcom 38 . . . . . . . 8 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦)))
8584ex 413 . . . . . . 7 (𝐹 ⊆ (𝐴 × 𝐵) → ((𝑥𝐹𝑦𝐹) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦))))
8685com23 86 . . . . . 6 (𝐹 ⊆ (𝐴 × 𝐵) → (𝐹:𝐴1-1𝐵 → ((𝑥𝐹𝑦𝐹) → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦))))
877, 86mpcom 38 . . . . 5 (𝐹:𝐴1-1𝐵 → ((𝑥𝐹𝑦𝐹) → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦)))
8887ralrimivv 3195 . . . 4 (𝐹:𝐴1-1𝐵 → ∀𝑥𝐹𝑦𝐹 (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦))
89 dff13 7202 . . . 4 ((2nd𝐹):𝐹1-1𝐵 ↔ ((2nd𝐹):𝐹𝐵 ∧ ∀𝑥𝐹𝑦𝐹 (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦)))
905, 88, 89sylanbrc 583 . . 3 (𝐹:𝐴1-1𝐵 → (2nd𝐹):𝐹1-1𝐵)
91 df-f1 6501 . . . 4 ((2nd𝐹):𝐹1-1𝐵 ↔ ((2nd𝐹):𝐹𝐵 ∧ Fun (2nd𝐹)))
9291simprbi 497 . . 3 ((2nd𝐹):𝐹1-1𝐵 → Fun (2nd𝐹))
9390, 92syl 17 . 2 (𝐹:𝐴1-1𝐵 → Fun (2nd𝐹))
94 dff1o3 6790 . 2 ((2nd𝐹):𝐹1-1-onto→ran 𝐹 ↔ ((2nd𝐹):𝐹onto→ran 𝐹 ∧ Fun (2nd𝐹)))
953, 93, 94sylanbrc 583 1 (𝐹:𝐴1-1𝐵 → (2nd𝐹):𝐹1-1-onto→ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  wrex 3073  wss 3910  cop 4592   × cxp 5631  ccnv 5632  ran crn 5634  cres 5635  Fun wfun 6490  wf 6492  1-1wf1 6493  ontowfo 6494  1-1-ontowf1o 6495  cfv 6496  2nd c2nd 7920
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-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  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-ral 3065  df-rex 3074  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-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  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-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-2nd 7922
This theorem is referenced by:  hashf1rn  14252
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