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Theorem f1o2ndf1 7625
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 6406 . . 3 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
2 fo2ndf 7624 . . 3 (𝐹:𝐴𝐵 → (2nd𝐹):𝐹onto→ran 𝐹)
31, 2syl 17 . 2 (𝐹:𝐴1-1𝐵 → (2nd𝐹):𝐹onto→ran 𝐹)
4 f2ndf 7623 . . . . 5 (𝐹:𝐴𝐵 → (2nd𝐹):𝐹𝐵)
51, 4syl 17 . . . 4 (𝐹:𝐴1-1𝐵 → (2nd𝐹):𝐹𝐵)
6 fssxp 6365 . . . . . . 7 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
71, 6syl 17 . . . . . 6 (𝐹:𝐴1-1𝐵𝐹 ⊆ (𝐴 × 𝐵))
8 ssel2 3855 . . . . . . . . . . 11 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ 𝑥𝐹) → 𝑥 ∈ (𝐴 × 𝐵))
9 elxp2 5432 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 × 𝐵) ↔ ∃𝑎𝐴𝑣𝐵 𝑥 = ⟨𝑎, 𝑣⟩)
108, 9sylib 210 . . . . . . . . . 10 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ 𝑥𝐹) → ∃𝑎𝐴𝑣𝐵 𝑥 = ⟨𝑎, 𝑣⟩)
11 ssel2 3855 . . . . . . . . . . 11 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ 𝑦𝐹) → 𝑦 ∈ (𝐴 × 𝐵))
12 elxp2 5432 . . . . . . . . . . 11 (𝑦 ∈ (𝐴 × 𝐵) ↔ ∃𝑏𝐴𝑤𝐵 𝑦 = ⟨𝑏, 𝑤⟩)
1311, 12sylib 210 . . . . . . . . . 10 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ 𝑦𝐹) → ∃𝑏𝐴𝑤𝐵 𝑦 = ⟨𝑏, 𝑤⟩)
1410, 13anim12dan 609 . . . . . . . . 9 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) → (∃𝑎𝐴𝑣𝐵 𝑥 = ⟨𝑎, 𝑣⟩ ∧ ∃𝑏𝐴𝑤𝐵 𝑦 = ⟨𝑏, 𝑤⟩))
15 fvres 6520 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨𝑎, 𝑣⟩ ∈ 𝐹 → ((2nd𝐹)‘⟨𝑎, 𝑣⟩) = (2nd ‘⟨𝑎, 𝑣⟩))
1615ad2antrr 713 . . . . . . . . . . . . . . . . . . . . . 22 (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → ((2nd𝐹)‘⟨𝑎, 𝑣⟩) = (2nd ‘⟨𝑎, 𝑣⟩))
17 fvres 6520 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨𝑏, 𝑤⟩ ∈ 𝐹 → ((2nd𝐹)‘⟨𝑏, 𝑤⟩) = (2nd ‘⟨𝑏, 𝑤⟩))
1817ad2antlr 714 . . . . . . . . . . . . . . . . . . . . . 22 (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → ((2nd𝐹)‘⟨𝑏, 𝑤⟩) = (2nd ‘⟨𝑏, 𝑤⟩))
1916, 18eqeq12d 2793 . . . . . . . . . . . . . . . . . . . . 21 (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → (((2nd𝐹)‘⟨𝑎, 𝑣⟩) = ((2nd𝐹)‘⟨𝑏, 𝑤⟩) ↔ (2nd ‘⟨𝑎, 𝑣⟩) = (2nd ‘⟨𝑏, 𝑤⟩)))
20 vex 3418 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑎 ∈ V
21 vex 3418 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑣 ∈ V
2220, 21op2nd 7512 . . . . . . . . . . . . . . . . . . . . . . 23 (2nd ‘⟨𝑎, 𝑣⟩) = 𝑣
23 vex 3418 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑏 ∈ V
24 vex 3418 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑤 ∈ V
2523, 24op2nd 7512 . . . . . . . . . . . . . . . . . . . . . . 23 (2nd ‘⟨𝑏, 𝑤⟩) = 𝑤
2622, 25eqeq12i 2792 . . . . . . . . . . . . . . . . . . . . . 22 ((2nd ‘⟨𝑎, 𝑣⟩) = (2nd ‘⟨𝑏, 𝑤⟩) ↔ 𝑣 = 𝑤)
27 f1fun 6408 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐹:𝐴1-1𝐵 → Fun 𝐹)
28 funopfv 6549 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Fun 𝐹 → (⟨𝑎, 𝑣⟩ ∈ 𝐹 → (𝐹𝑎) = 𝑣))
29 funopfv 6549 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Fun 𝐹 → (⟨𝑏, 𝑤⟩ ∈ 𝐹 → (𝐹𝑏) = 𝑤))
3028, 29anim12d 599 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2785 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐹𝑎) = 𝑣𝑣 = (𝐹𝑎))
3332biimpi 208 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐹𝑎) = 𝑣𝑣 = (𝐹𝑎))
34 eqcom 2785 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐹𝑏) = 𝑤𝑤 = (𝐹𝑏))
3534biimpi 208 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐹𝑏) = 𝑤𝑤 = (𝐹𝑏))
3633, 35eqeqan12d 2794 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝐹𝑎) = 𝑣 ∧ (𝐹𝑏) = 𝑤) → (𝑣 = 𝑤 ↔ (𝐹𝑎) = (𝐹𝑏)))
37 simpl 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑎𝐴𝑣𝐵) → 𝑎𝐴)
38 simpl 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑏𝐴𝑤𝐵) → 𝑏𝐴)
3937, 38anim12i 603 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → (𝑎𝐴𝑏𝐴))
40 f1veqaeq 6842 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝐹:𝐴1-1𝐵 ∧ (𝑎𝐴𝑏𝐴)) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
4139, 40sylan2 583 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝐹:𝐴1-1𝐵 ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
42 opeq12 4680 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑎 = 𝑏𝑣 = 𝑤) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)
4342ex 405 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 405 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 245 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 399 . . . . . . . . . . . . . . . . . . . . . . 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, 56syl5bi 234 . . . . . . . . . . . . . . . . . . . . 21 (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → ((2nd ‘⟨𝑎, 𝑣⟩) = (2nd ‘⟨𝑏, 𝑤⟩) → (𝐹:𝐴1-1𝐵 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)))
5819, 57sylbid 232 . . . . . . . . . . . . . . . . . . . 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 405 . . . . . . . . . . . . . . . . . 18 ((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) → (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘⟨𝑎, 𝑣⟩) = ((2nd𝐹)‘⟨𝑏, 𝑤⟩) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
6160adantl 474 . . . . . . . . . . . . . . . . 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 738 . . . . . . . . . . . . . . 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 2853 . . . . . . . . . . . . . . . . . 18 (𝑥 = ⟨𝑎, 𝑣⟩ → (𝑥𝐹 ↔ ⟨𝑎, 𝑣⟩ ∈ 𝐹))
6564ad2antlr 714 . . . . . . . . . . . . . . . . 17 ((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) → (𝑥𝐹 ↔ ⟨𝑎, 𝑣⟩ ∈ 𝐹))
66 eleq1 2853 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑏, 𝑤⟩ → (𝑦𝐹 ↔ ⟨𝑏, 𝑤⟩ ∈ 𝐹))
6765, 66bi2anan9 626 . . . . . . . . . . . . . . . 16 (((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → ((𝑥𝐹𝑦𝐹) ↔ (⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹)))
6867anbi2d 619 . . . . . . . . . . . . . . 15 (((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ (⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹))))
69 fveq2 6501 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ⟨𝑎, 𝑣⟩ → ((2nd𝐹)‘𝑥) = ((2nd𝐹)‘⟨𝑎, 𝑣⟩))
7069ad2antlr 714 . . . . . . . . . . . . . . . . . 18 ((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) → ((2nd𝐹)‘𝑥) = ((2nd𝐹)‘⟨𝑎, 𝑣⟩))
71 fveq2 6501 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨𝑏, 𝑤⟩ → ((2nd𝐹)‘𝑦) = ((2nd𝐹)‘⟨𝑏, 𝑤⟩))
7270, 71eqeqan12d 2794 . . . . . . . . . . . . . . . . 17 (((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) ↔ ((2nd𝐹)‘⟨𝑎, 𝑣⟩) = ((2nd𝐹)‘⟨𝑏, 𝑤⟩)))
73 simpllr 763 . . . . . . . . . . . . . . . . . 18 (((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → 𝑥 = ⟨𝑎, 𝑣⟩)
74 simpr 477 . . . . . . . . . . . . . . . . . 18 (((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → 𝑦 = ⟨𝑏, 𝑤⟩)
7573, 74eqeq12d 2793 . . . . . . . . . . . . . . . . 17 (((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → (𝑥 = 𝑦 ↔ ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))
7672, 75imbi12d 337 . . . . . . . . . . . . . . . 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 333 . . . . . . . . . . . . . . 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 286 . . . . . . . . . . . . . 14 (((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦))))
7978ex 405 . . . . . . . . . . . . 13 ((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) → (𝑦 = ⟨𝑏, 𝑤⟩ → ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦)))))
8079rexlimdvva 3239 . . . . . . . . . . . 12 (((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) → (∃𝑏𝐴𝑤𝐵 𝑦 = ⟨𝑏, 𝑤⟩ → ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦)))))
8180ex 405 . . . . . . . . . . 11 ((𝑎𝐴𝑣𝐵) → (𝑥 = ⟨𝑎, 𝑣⟩ → (∃𝑏𝐴𝑤𝐵 𝑦 = ⟨𝑏, 𝑤⟩ → ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦))))))
8281rexlimivv 3237 . . . . . . . . . 10 (∃𝑎𝐴𝑣𝐵 𝑥 = ⟨𝑎, 𝑣⟩ → (∃𝑏𝐴𝑤𝐵 𝑦 = ⟨𝑏, 𝑤⟩ → ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦)))))
8382imp 398 . . . . . . . . 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 405 . . . . . . 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 3140 . . . 4 (𝐹:𝐴1-1𝐵 → ∀𝑥𝐹𝑦𝐹 (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦))
89 dff13 6840 . . . 4 ((2nd𝐹):𝐹1-1𝐵 ↔ ((2nd𝐹):𝐹𝐵 ∧ ∀𝑥𝐹𝑦𝐹 (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦)))
905, 88, 89sylanbrc 575 . . 3 (𝐹:𝐴1-1𝐵 → (2nd𝐹):𝐹1-1𝐵)
91 df-f1 6195 . . . 4 ((2nd𝐹):𝐹1-1𝐵 ↔ ((2nd𝐹):𝐹𝐵 ∧ Fun (2nd𝐹)))
9291simprbi 489 . . 3 ((2nd𝐹):𝐹1-1𝐵 → Fun (2nd𝐹))
9390, 92syl 17 . 2 (𝐹:𝐴1-1𝐵 → Fun (2nd𝐹))
94 dff1o3 6452 . 2 ((2nd𝐹):𝐹1-1-onto→ran 𝐹 ↔ ((2nd𝐹):𝐹onto→ran 𝐹 ∧ Fun (2nd𝐹)))
953, 93, 94sylanbrc 575 1 (𝐹:𝐴1-1𝐵 → (2nd𝐹):𝐹1-1-onto→ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2050  wral 3088  wrex 3089  wss 3831  cop 4448   × cxp 5406  ccnv 5407  ran crn 5409  cres 5410  Fun wfun 6184  wf 6186  1-1wf1 6187  ontowfo 6188  1-1-ontowf1o 6189  cfv 6190  2nd c2nd 7502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-id 5313  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-2nd 7504
This theorem is referenced by:  hashf1rn  13531
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