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Theorem f1o2ndf1 6093
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 5298 . . 3 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
2 fo2ndf 6092 . . 3 (𝐹:𝐴𝐵 → (2nd𝐹):𝐹onto→ran 𝐹)
31, 2syl 14 . 2 (𝐹:𝐴1-1𝐵 → (2nd𝐹):𝐹onto→ran 𝐹)
4 f2ndf 6091 . . . . 5 (𝐹:𝐴𝐵 → (2nd𝐹):𝐹𝐵)
51, 4syl 14 . . . 4 (𝐹:𝐴1-1𝐵 → (2nd𝐹):𝐹𝐵)
6 fssxp 5260 . . . . . . 7 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
71, 6syl 14 . . . . . 6 (𝐹:𝐴1-1𝐵𝐹 ⊆ (𝐴 × 𝐵))
8 ssel2 3062 . . . . . . . . . . 11 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ 𝑥𝐹) → 𝑥 ∈ (𝐴 × 𝐵))
9 elxp2 4527 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 × 𝐵) ↔ ∃𝑎𝐴𝑣𝐵 𝑥 = ⟨𝑎, 𝑣⟩)
108, 9sylib 121 . . . . . . . . . 10 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ 𝑥𝐹) → ∃𝑎𝐴𝑣𝐵 𝑥 = ⟨𝑎, 𝑣⟩)
11 ssel2 3062 . . . . . . . . . . 11 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ 𝑦𝐹) → 𝑦 ∈ (𝐴 × 𝐵))
12 elxp2 4527 . . . . . . . . . . 11 (𝑦 ∈ (𝐴 × 𝐵) ↔ ∃𝑏𝐴𝑤𝐵 𝑦 = ⟨𝑏, 𝑤⟩)
1311, 12sylib 121 . . . . . . . . . 10 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ 𝑦𝐹) → ∃𝑏𝐴𝑤𝐵 𝑦 = ⟨𝑏, 𝑤⟩)
1410, 13anim12dan 574 . . . . . . . . 9 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) → (∃𝑎𝐴𝑣𝐵 𝑥 = ⟨𝑎, 𝑣⟩ ∧ ∃𝑏𝐴𝑤𝐵 𝑦 = ⟨𝑏, 𝑤⟩))
15 fvres 5413 . . . . . . . . . . . . . . . . . . . . . . . . 25 (⟨𝑎, 𝑣⟩ ∈ 𝐹 → ((2nd𝐹)‘⟨𝑎, 𝑣⟩) = (2nd ‘⟨𝑎, 𝑣⟩))
1615adantr 274 . . . . . . . . . . . . . . . . . . . . . . . 24 ((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) → ((2nd𝐹)‘⟨𝑎, 𝑣⟩) = (2nd ‘⟨𝑎, 𝑣⟩))
1716adantr 274 . . . . . . . . . . . . . . . . . . . . . . 23 (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → ((2nd𝐹)‘⟨𝑎, 𝑣⟩) = (2nd ‘⟨𝑎, 𝑣⟩))
18 fvres 5413 . . . . . . . . . . . . . . . . . . . . . . . 24 (⟨𝑏, 𝑤⟩ ∈ 𝐹 → ((2nd𝐹)‘⟨𝑏, 𝑤⟩) = (2nd ‘⟨𝑏, 𝑤⟩))
1918ad2antlr 480 . . . . . . . . . . . . . . . . . . . . . . 23 (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → ((2nd𝐹)‘⟨𝑏, 𝑤⟩) = (2nd ‘⟨𝑏, 𝑤⟩))
2017, 19eqeq12d 2132 . . . . . . . . . . . . . . . . . . . . . 22 (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → (((2nd𝐹)‘⟨𝑎, 𝑣⟩) = ((2nd𝐹)‘⟨𝑏, 𝑤⟩) ↔ (2nd ‘⟨𝑎, 𝑣⟩) = (2nd ‘⟨𝑏, 𝑤⟩)))
21 vex 2663 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑎 ∈ V
22 vex 2663 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑣 ∈ V
2321, 22op2nd 6013 . . . . . . . . . . . . . . . . . . . . . . . 24 (2nd ‘⟨𝑎, 𝑣⟩) = 𝑣
24 vex 2663 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑏 ∈ V
25 vex 2663 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑤 ∈ V
2624, 25op2nd 6013 . . . . . . . . . . . . . . . . . . . . . . . 24 (2nd ‘⟨𝑏, 𝑤⟩) = 𝑤
2723, 26eqeq12i 2131 . . . . . . . . . . . . . . . . . . . . . . 23 ((2nd ‘⟨𝑎, 𝑣⟩) = (2nd ‘⟨𝑏, 𝑤⟩) ↔ 𝑣 = 𝑤)
28 f1fun 5301 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐹:𝐴1-1𝐵 → Fun 𝐹)
29 funopfv 5429 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Fun 𝐹 → (⟨𝑎, 𝑣⟩ ∈ 𝐹 → (𝐹𝑎) = 𝑣))
30 funopfv 5429 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Fun 𝐹 → (⟨𝑏, 𝑤⟩ ∈ 𝐹 → (𝐹𝑏) = 𝑤))
3129, 30anim12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2119 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝐹𝑎) = 𝑣𝑣 = (𝐹𝑎))
3433biimpi 119 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐹𝑎) = 𝑣𝑣 = (𝐹𝑎))
35 eqcom 2119 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝐹𝑏) = 𝑤𝑤 = (𝐹𝑏))
3635biimpi 119 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐹𝑏) = 𝑤𝑤 = (𝐹𝑏))
3734, 36eqeqan12d 2133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝐹𝑎) = 𝑣 ∧ (𝐹𝑏) = 𝑤) → (𝑣 = 𝑤 ↔ (𝐹𝑎) = (𝐹𝑏)))
38 simpl 108 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎𝐴𝑣𝐵) → 𝑎𝐴)
39 simpl 108 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑏𝐴𝑤𝐵) → 𝑏𝐴)
4038, 39anim12i 336 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → (𝑎𝐴𝑏𝐴))
41 f1veqaeq 5638 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝐹:𝐴1-1𝐵 ∧ (𝑎𝐴𝑏𝐴)) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
4240, 41sylan2 284 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝐹:𝐴1-1𝐵 ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
43 opeq12 3677 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑎 = 𝑏𝑣 = 𝑤) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)
4443ex 114 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑎 = 𝑏 → (𝑣 = 𝑤 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))
4542, 44syl6 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝐹:𝐴1-1𝐵 ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → ((𝐹𝑎) = (𝐹𝑏) → (𝑣 = 𝑤 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)))
4645com23 78 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝐹:𝐴1-1𝐵 ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → (𝑣 = 𝑤 → ((𝐹𝑎) = (𝐹𝑏) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)))
4746ex 114 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝐹:𝐴1-1𝐵 → (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → (𝑣 = 𝑤 → ((𝐹𝑎) = (𝐹𝑏) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
4847com14 88 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐹𝑎) = (𝐹𝑏) → (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → (𝑣 = 𝑤 → (𝐹:𝐴1-1𝐵 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
4937, 48syl6bi 162 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝐹𝑎) = 𝑣 ∧ (𝐹𝑏) = 𝑤) → (𝑣 = 𝑤 → (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → (𝑣 = 𝑤 → (𝐹:𝐴1-1𝐵 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)))))
5049com14 88 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑣 = 𝑤 → (𝑣 = 𝑤 → (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → (((𝐹𝑎) = 𝑣 ∧ (𝐹𝑏) = 𝑤) → (𝐹:𝐴1-1𝐵 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)))))
5150pm2.43i 49 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑣 = 𝑤 → (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → (((𝐹𝑎) = 𝑣 ∧ (𝐹𝑏) = 𝑤) → (𝐹:𝐴1-1𝐵 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
5251com14 88 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐹:𝐴1-1𝐵 → (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → (((𝐹𝑎) = 𝑣 ∧ (𝐹𝑏) = 𝑤) → (𝑣 = 𝑤 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
5352com23 78 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐹:𝐴1-1𝐵 → (((𝐹𝑎) = 𝑣 ∧ (𝐹𝑏) = 𝑤) → (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → (𝑣 = 𝑤 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
5432, 53syld 45 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐹:𝐴1-1𝐵 → ((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) → (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → (𝑣 = 𝑤 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
5554com13 80 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → ((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) → (𝐹:𝐴1-1𝐵 → (𝑣 = 𝑤 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
5655impcom 124 . . . . . . . . . . . . . . . . . . . . . . . 24 (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → (𝐹:𝐴1-1𝐵 → (𝑣 = 𝑤 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)))
5756com23 78 . . . . . . . . . . . . . . . . . . . . . . 23 (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → (𝑣 = 𝑤 → (𝐹:𝐴1-1𝐵 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)))
5827, 57syl5bi 151 . . . . . . . . . . . . . . . . . . . . . 22 (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → ((2nd ‘⟨𝑎, 𝑣⟩) = (2nd ‘⟨𝑏, 𝑤⟩) → (𝐹:𝐴1-1𝐵 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)))
5920, 58sylbid 149 . . . . . . . . . . . . . . . . . . . . 21 (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → (((2nd𝐹)‘⟨𝑎, 𝑣⟩) = ((2nd𝐹)‘⟨𝑏, 𝑤⟩) → (𝐹:𝐴1-1𝐵 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)))
6059com23 78 . . . . . . . . . . . . . . . . . . . 20 (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵))) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘⟨𝑎, 𝑣⟩) = ((2nd𝐹)‘⟨𝑏, 𝑤⟩) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)))
6160ex 114 . . . . . . . . . . . . . . . . . . 19 ((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) → (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘⟨𝑎, 𝑣⟩) = ((2nd𝐹)‘⟨𝑏, 𝑤⟩) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
6261adantl 275 . . . . . . . . . . . . . . . . . 18 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹)) → (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘⟨𝑎, 𝑣⟩) = ((2nd𝐹)‘⟨𝑏, 𝑤⟩) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
6362com12 30 . . . . . . . . . . . . . . . . 17 (((𝑎𝐴𝑣𝐵) ∧ (𝑏𝐴𝑤𝐵)) → ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘⟨𝑎, 𝑣⟩) = ((2nd𝐹)‘⟨𝑏, 𝑤⟩) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
6463adantlr 468 . . . . . . . . . . . . . . . 16 ((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) → ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘⟨𝑎, 𝑣⟩) = ((2nd𝐹)‘⟨𝑏, 𝑤⟩) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
6564adantr 274 . . . . . . . . . . . . . . 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⟩))))
66 eleq1 2180 . . . . . . . . . . . . . . . . . 18 (𝑥 = ⟨𝑎, 𝑣⟩ → (𝑥𝐹 ↔ ⟨𝑎, 𝑣⟩ ∈ 𝐹))
6766ad2antlr 480 . . . . . . . . . . . . . . . . 17 ((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) → (𝑥𝐹 ↔ ⟨𝑎, 𝑣⟩ ∈ 𝐹))
68 eleq1 2180 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑏, 𝑤⟩ → (𝑦𝐹 ↔ ⟨𝑏, 𝑤⟩ ∈ 𝐹))
6967, 68bi2anan9 580 . . . . . . . . . . . . . . . 16 (((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → ((𝑥𝐹𝑦𝐹) ↔ (⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹)))
7069anbi2d 459 . . . . . . . . . . . . . . 15 (((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ (⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹))))
71 fveq2 5389 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ⟨𝑎, 𝑣⟩ → ((2nd𝐹)‘𝑥) = ((2nd𝐹)‘⟨𝑎, 𝑣⟩))
7271ad2antlr 480 . . . . . . . . . . . . . . . . . 18 ((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) → ((2nd𝐹)‘𝑥) = ((2nd𝐹)‘⟨𝑎, 𝑣⟩))
73 fveq2 5389 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨𝑏, 𝑤⟩ → ((2nd𝐹)‘𝑦) = ((2nd𝐹)‘⟨𝑏, 𝑤⟩))
7472, 73eqeqan12d 2133 . . . . . . . . . . . . . . . . 17 (((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) ↔ ((2nd𝐹)‘⟨𝑎, 𝑣⟩) = ((2nd𝐹)‘⟨𝑏, 𝑤⟩)))
75 simpllr 508 . . . . . . . . . . . . . . . . . 18 (((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → 𝑥 = ⟨𝑎, 𝑣⟩)
76 simpr 109 . . . . . . . . . . . . . . . . . 18 (((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → 𝑦 = ⟨𝑏, 𝑤⟩)
7775, 76eqeq12d 2132 . . . . . . . . . . . . . . . . 17 (((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → (𝑥 = 𝑦 ↔ ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))
7874, 77imbi12d 233 . . . . . . . . . . . . . . . 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⟩)))
7978imbi2d 229 . . . . . . . . . . . . . . 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⟩))))
8065, 70, 793imtr4d 202 . . . . . . . . . . . . . 14 (((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦))))
8180ex 114 . . . . . . . . . . . . 13 ((((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏𝐴𝑤𝐵)) → (𝑦 = ⟨𝑏, 𝑤⟩ → ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦)))))
8281rexlimdvva 2534 . . . . . . . . . . . 12 (((𝑎𝐴𝑣𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) → (∃𝑏𝐴𝑤𝐵 𝑦 = ⟨𝑏, 𝑤⟩ → ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦)))))
8382ex 114 . . . . . . . . . . 11 ((𝑎𝐴𝑣𝐵) → (𝑥 = ⟨𝑎, 𝑣⟩ → (∃𝑏𝐴𝑤𝐵 𝑦 = ⟨𝑏, 𝑤⟩ → ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦))))))
8483rexlimivv 2532 . . . . . . . . . 10 (∃𝑎𝐴𝑣𝐵 𝑥 = ⟨𝑎, 𝑣⟩ → (∃𝑏𝐴𝑤𝐵 𝑦 = ⟨𝑏, 𝑤⟩ → ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦)))))
8584imp 123 . . . . . . . . 9 ((∃𝑎𝐴𝑣𝐵 𝑥 = ⟨𝑎, 𝑣⟩ ∧ ∃𝑏𝐴𝑤𝐵 𝑦 = ⟨𝑏, 𝑤⟩) → ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦))))
8614, 85mpcom 36 . . . . . . . 8 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐹𝑦𝐹)) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦)))
8786ex 114 . . . . . . 7 (𝐹 ⊆ (𝐴 × 𝐵) → ((𝑥𝐹𝑦𝐹) → (𝐹:𝐴1-1𝐵 → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦))))
8887com23 78 . . . . . 6 (𝐹 ⊆ (𝐴 × 𝐵) → (𝐹:𝐴1-1𝐵 → ((𝑥𝐹𝑦𝐹) → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦))))
897, 88mpcom 36 . . . . 5 (𝐹:𝐴1-1𝐵 → ((𝑥𝐹𝑦𝐹) → (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦)))
9089ralrimivv 2490 . . . 4 (𝐹:𝐴1-1𝐵 → ∀𝑥𝐹𝑦𝐹 (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦))
91 dff13 5637 . . . 4 ((2nd𝐹):𝐹1-1𝐵 ↔ ((2nd𝐹):𝐹𝐵 ∧ ∀𝑥𝐹𝑦𝐹 (((2nd𝐹)‘𝑥) = ((2nd𝐹)‘𝑦) → 𝑥 = 𝑦)))
925, 90, 91sylanbrc 413 . . 3 (𝐹:𝐴1-1𝐵 → (2nd𝐹):𝐹1-1𝐵)
93 df-f1 5098 . . . 4 ((2nd𝐹):𝐹1-1𝐵 ↔ ((2nd𝐹):𝐹𝐵 ∧ Fun (2nd𝐹)))
9493simprbi 273 . . 3 ((2nd𝐹):𝐹1-1𝐵 → Fun (2nd𝐹))
9592, 94syl 14 . 2 (𝐹:𝐴1-1𝐵 → Fun (2nd𝐹))
96 dff1o3 5341 . 2 ((2nd𝐹):𝐹1-1-onto→ran 𝐹 ↔ ((2nd𝐹):𝐹onto→ran 𝐹 ∧ Fun (2nd𝐹)))
973, 95, 96sylanbrc 413 1 (𝐹:𝐴1-1𝐵 → (2nd𝐹):𝐹1-1-onto→ran 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wcel 1465  wral 2393  wrex 2394  wss 3041  cop 3500   × cxp 4507  ccnv 4508  ran crn 4510  cres 4511  Fun wfun 5087  wf 5089  1-1wf1 5090  ontowfo 5091  1-1-ontowf1o 5092  cfv 5093  2nd c2nd 6005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-2nd 6007
This theorem is referenced by:  fihashf1rn  10503
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