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Theorem iccpnfcnv 24893
Description: Define a bijection from [0, 1] to [0, +∞]. (Contributed by Mario Carneiro, 9-Sep-2015.)
Hypothesis
Ref Expression
iccpnfhmeo.f 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))))
Assertion
Ref Expression
iccpnfcnv (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦)))))
Distinct variable groups:   𝑥,𝑦   𝑦,𝐹
Allowed substitution hint:   𝐹(𝑥)
Proof of Theorem iccpnfcnv
StepHypRef Expression
1 iccpnfhmeo.f . . 3 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))))
2 0xr 11282 . . . . . . 7 0 ∈ ℝ*
3 pnfxr 11289 . . . . . . 7 +∞ ∈ ℝ*
4 0lepnf 13149 . . . . . . 7 0 ≤ +∞
5 ubicc2 13482 . . . . . . 7 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞))
62, 3, 4, 5mp3an 1463 . . . . . 6 +∞ ∈ (0[,]+∞)
76a1i 11 . . . . 5 ((𝑥 ∈ (0[,]1) ∧ 𝑥 = 1) → +∞ ∈ (0[,]+∞))
8 icossicc 13453 . . . . . 6 (0[,)+∞) ⊆ (0[,]+∞)
9 1xr 11294 . . . . . . . . . . . . . 14 1 ∈ ℝ*
10 0le1 11760 . . . . . . . . . . . . . 14 0 ≤ 1
11 snunico 13496 . . . . . . . . . . . . . 14 ((0 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ 0 ≤ 1) → ((0[,)1) ∪ {1}) = (0[,]1))
122, 9, 10, 11mp3an 1463 . . . . . . . . . . . . 13 ((0[,)1) ∪ {1}) = (0[,]1)
1312eleq2i 2826 . . . . . . . . . . . 12 (𝑥 ∈ ((0[,)1) ∪ {1}) ↔ 𝑥 ∈ (0[,]1))
14 elun 4128 . . . . . . . . . . . 12 (𝑥 ∈ ((0[,)1) ∪ {1}) ↔ (𝑥 ∈ (0[,)1) ∨ 𝑥 ∈ {1}))
1513, 14bitr3i 277 . . . . . . . . . . 11 (𝑥 ∈ (0[,]1) ↔ (𝑥 ∈ (0[,)1) ∨ 𝑥 ∈ {1}))
16 pm2.53 851 . . . . . . . . . . 11 ((𝑥 ∈ (0[,)1) ∨ 𝑥 ∈ {1}) → (¬ 𝑥 ∈ (0[,)1) → 𝑥 ∈ {1}))
1715, 16sylbi 217 . . . . . . . . . 10 (𝑥 ∈ (0[,]1) → (¬ 𝑥 ∈ (0[,)1) → 𝑥 ∈ {1}))
18 elsni 4618 . . . . . . . . . 10 (𝑥 ∈ {1} → 𝑥 = 1)
1917, 18syl6 35 . . . . . . . . 9 (𝑥 ∈ (0[,]1) → (¬ 𝑥 ∈ (0[,)1) → 𝑥 = 1))
2019con1d 145 . . . . . . . 8 (𝑥 ∈ (0[,]1) → (¬ 𝑥 = 1 → 𝑥 ∈ (0[,)1)))
2120imp 406 . . . . . . 7 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 1) → 𝑥 ∈ (0[,)1))
22 eqid 2735 . . . . . . . . . . . 12 (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) = (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))
2322icopnfcnv 24891 . . . . . . . . . . 11 ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))):(0[,)1)–1-1-onto→(0[,)+∞) ∧ ◡(𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) = (𝑦 ∈ (0[,)+∞) ↦ (𝑦 / (1 + 𝑦))))
2423simpli 483 . . . . . . . . . 10 (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))):(0[,)1)–1-1-onto→(0[,)+∞)
25 f1of 6818 . . . . . . . . . 10 ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))):(0[,)1)–1-1-onto→(0[,)+∞) → (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))):(0[,)1)⟶(0[,)+∞))
2624, 25ax-mp 5 . . . . . . . . 9 (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))):(0[,)1)⟶(0[,)+∞)
2722fmpt 7100 . . . . . . . . 9 (∀𝑥 ∈ (0[,)1)(𝑥 / (1 − 𝑥)) ∈ (0[,)+∞) ↔ (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))):(0[,)1)⟶(0[,)+∞))
2826, 27mpbir 231 . . . . . . . 8 ∀𝑥 ∈ (0[,)1)(𝑥 / (1 − 𝑥)) ∈ (0[,)+∞)
2928rspec 3233 . . . . . . 7 (𝑥 ∈ (0[,)1) → (𝑥 / (1 − 𝑥)) ∈ (0[,)+∞))
3021, 29syl 17 . . . . . 6 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 1) → (𝑥 / (1 − 𝑥)) ∈ (0[,)+∞))
318, 30sselid 3956 . . . . 5 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 1) → (𝑥 / (1 − 𝑥)) ∈ (0[,]+∞))
327, 31ifclda 4536 . . . 4 (𝑥 ∈ (0[,]1) → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ∈ (0[,]+∞))
3332adantl 481 . . 3 ((⊤ ∧ 𝑥 ∈ (0[,]1)) → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ∈ (0[,]+∞))
34 1elunit 13487 . . . . . 6 1 ∈ (0[,]1)
3534a1i 11 . . . . 5 ((𝑦 ∈ (0[,]+∞) ∧ 𝑦 = +∞) → 1 ∈ (0[,]1))
36 icossicc 13453 . . . . . 6 (0[,)1) ⊆ (0[,]1)
37 snunico 13496 . . . . . . . . . . . . . 14 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → ((0[,)+∞) ∪ {+∞}) = (0[,]+∞))
382, 3, 4, 37mp3an 1463 . . . . . . . . . . . . 13 ((0[,)+∞) ∪ {+∞}) = (0[,]+∞)
3938eleq2i 2826 . . . . . . . . . . . 12 (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ 𝑦 ∈ (0[,]+∞))
40 elun 4128 . . . . . . . . . . . 12 (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
4139, 40bitr3i 277 . . . . . . . . . . 11 (𝑦 ∈ (0[,]+∞) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
42 pm2.53 851 . . . . . . . . . . 11 ((𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
4341, 42sylbi 217 . . . . . . . . . 10 (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
44 elsni 4618 . . . . . . . . . 10 (𝑦 ∈ {+∞} → 𝑦 = +∞)
4543, 44syl6 35 . . . . . . . . 9 (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 = +∞))
4645con1d 145 . . . . . . . 8 (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 = +∞ → 𝑦 ∈ (0[,)+∞)))
4746imp 406 . . . . . . 7 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 𝑦 ∈ (0[,)+∞))
48 f1ocnv 6830 . . . . . . . . . 10 ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))):(0[,)1)–1-1-onto→(0[,)+∞) → ◡(𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))):(0[,)+∞)–1-1-onto→(0[,)1))
49 f1of 6818 . . . . . . . . . 10 (◡(𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))):(0[,)+∞)–1-1-onto→(0[,)1) → ◡(𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))):(0[,)+∞)⟶(0[,)1))
5024, 48, 49mp2b 10 . . . . . . . . 9 ◡(𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))):(0[,)+∞)⟶(0[,)1)
5123simpri 485 . . . . . . . . . 10 ◡(𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) = (𝑦 ∈ (0[,)+∞) ↦ (𝑦 / (1 + 𝑦)))
5251fmpt 7100 . . . . . . . . 9 (∀𝑦 ∈ (0[,)+∞)(𝑦 / (1 + 𝑦)) ∈ (0[,)1) ↔ ◡(𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))):(0[,)+∞)⟶(0[,)1))
5350, 52mpbir 231 . . . . . . . 8 ∀𝑦 ∈ (0[,)+∞)(𝑦 / (1 + 𝑦)) ∈ (0[,)1)
5453rspec 3233 . . . . . . 7 (𝑦 ∈ (0[,)+∞) → (𝑦 / (1 + 𝑦)) ∈ (0[,)1))
5547, 54syl 17 . . . . . 6 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (𝑦 / (1 + 𝑦)) ∈ (0[,)1))
5636, 55sselid 3956 . . . . 5 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (𝑦 / (1 + 𝑦)) ∈ (0[,]1))
5735, 56ifclda 4536 . . . 4 (𝑦 ∈ (0[,]+∞) → if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))) ∈ (0[,]1))
5857adantl 481 . . 3 ((⊤ ∧ 𝑦 ∈ (0[,]+∞)) → if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))) ∈ (0[,]1))
59 eqeq2 2747 . . . . . 6 (1 = if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))) → (𝑥 = 1 ↔ 𝑥 = if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦)))))
6059bibi1d 343 . . . . 5 (1 = if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))) → ((𝑥 = 1 ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))) ↔ (𝑥 = if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))) ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))))))
61 eqeq2 2747 . . . . . 6 ((𝑦 / (1 + 𝑦)) = if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))) → (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑥 = if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦)))))
6261bibi1d 343 . . . . 5 ((𝑦 / (1 + 𝑦)) = if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))) → ((𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))) ↔ (𝑥 = if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))) ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))))))
63 simpr 484 . . . . . . 7 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → 𝑦 = +∞)
64 iftrue 4506 . . . . . . . 8 (𝑥 = 1 → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) = +∞)
6564eqeq2d 2746 . . . . . . 7 (𝑥 = 1 → (𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ↔ 𝑦 = +∞))
6663, 65syl5ibrcom 247 . . . . . 6 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑥 = 1 → 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))))
67 pnfnre 11276 . . . . . . . . 9 +∞ ∉ ℝ
68 neleq1 3042 . . . . . . . . . 10 (𝑦 = +∞ → (𝑦 ∉ ℝ ↔ +∞ ∉ ℝ))
6968adantl 481 . . . . . . . . 9 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑦 ∉ ℝ ↔ +∞ ∉ ℝ))
7067, 69mpbiri 258 . . . . . . . 8 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → 𝑦 ∉ ℝ)
71 neleq1 3042 . . . . . . . 8 (𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) → (𝑦 ∉ ℝ ↔ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ∉ ℝ))
7270, 71syl5ibcom 245 . . . . . . 7 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ∉ ℝ))
73 df-nel 3037 . . . . . . . 8 (if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ∉ ℝ ↔ ¬ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ∈ ℝ)
74 iffalse 4509 . . . . . . . . . . . . 13 (¬ 𝑥 = 1 → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) = (𝑥 / (1 − 𝑥)))
7574adantl 481 . . . . . . . . . . . 12 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 1) → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) = (𝑥 / (1 − 𝑥)))
76 rge0ssre 13473 . . . . . . . . . . . . 13 (0[,)+∞) ⊆ ℝ
7776, 30sselid 3956 . . . . . . . . . . . 12 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 1) → (𝑥 / (1 − 𝑥)) ∈ ℝ)
7875, 77eqeltrd 2834 . . . . . . . . . . 11 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 1) → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ∈ ℝ)
7978ex 412 . . . . . . . . . 10 (𝑥 ∈ (0[,]1) → (¬ 𝑥 = 1 → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ∈ ℝ))
8079ad2antrr 726 . . . . . . . . 9 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (¬ 𝑥 = 1 → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ∈ ℝ))
8180con1d 145 . . . . . . . 8 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (¬ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ∈ ℝ → 𝑥 = 1))
8273, 81biimtrid 242 . . . . . . 7 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ∉ ℝ → 𝑥 = 1))
8372, 82syld 47 . . . . . 6 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) → 𝑥 = 1))
8466, 83impbid 212 . . . . 5 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑥 = 1 ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))))
85 eqeq2 2747 . . . . . . 7 (+∞ = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) → (𝑦 = +∞ ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))))
8685bibi2d 342 . . . . . 6 (+∞ = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) → ((𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑦 = +∞) ↔ (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))))))
87 eqeq2 2747 . . . . . . 7 ((𝑥 / (1 − 𝑥)) = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) → (𝑦 = (𝑥 / (1 − 𝑥)) ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))))
8887bibi2d 342 . . . . . 6 ((𝑥 / (1 − 𝑥)) = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) → ((𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑦 = (𝑥 / (1 − 𝑥))) ↔ (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))))))
89 0re 11237 . . . . . . . . . . . . . . 15 0 ∈ ℝ
90 elico2 13427 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ ∧ 1 ∈ ℝ*) → ((𝑦 / (1 + 𝑦)) ∈ (0[,)1) ↔ ((𝑦 / (1 + 𝑦)) ∈ ℝ ∧ 0 ≤ (𝑦 / (1 + 𝑦)) ∧ (𝑦 / (1 + 𝑦)) < 1)))
9189, 9, 90mp2an 692 . . . . . . . . . . . . . 14 ((𝑦 / (1 + 𝑦)) ∈ (0[,)1) ↔ ((𝑦 / (1 + 𝑦)) ∈ ℝ ∧ 0 ≤ (𝑦 / (1 + 𝑦)) ∧ (𝑦 / (1 + 𝑦)) < 1))
9255, 91sylib 218 . . . . . . . . . . . . 13 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → ((𝑦 / (1 + 𝑦)) ∈ ℝ ∧ 0 ≤ (𝑦 / (1 + 𝑦)) ∧ (𝑦 / (1 + 𝑦)) < 1))
9392simp1d 1142 . . . . . . . . . . . 12 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (𝑦 / (1 + 𝑦)) ∈ ℝ)
9492simp3d 1144 . . . . . . . . . . . 12 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (𝑦 / (1 + 𝑦)) < 1)
9593, 94gtned 11370 . . . . . . . . . . 11 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 1 ≠ (𝑦 / (1 + 𝑦)))
9695adantll 714 . . . . . . . . . 10 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → 1 ≠ (𝑦 / (1 + 𝑦)))
9796neneqd 2937 . . . . . . . . 9 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → ¬ 1 = (𝑦 / (1 + 𝑦)))
98 eqeq1 2739 . . . . . . . . . 10 (𝑥 = 1 → (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 1 = (𝑦 / (1 + 𝑦))))
9998notbid 318 . . . . . . . . 9 (𝑥 = 1 → (¬ 𝑥 = (𝑦 / (1 + 𝑦)) ↔ ¬ 1 = (𝑦 / (1 + 𝑦))))
10097, 99syl5ibrcom 247 . . . . . . . 8 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → (𝑥 = 1 → ¬ 𝑥 = (𝑦 / (1 + 𝑦))))
101100imp 406 . . . . . . 7 ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 1) → ¬ 𝑥 = (𝑦 / (1 + 𝑦)))
102 simplr 768 . . . . . . 7 ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 1) → ¬ 𝑦 = +∞)
103101, 1022falsed 376 . . . . . 6 ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 1) → (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑦 = +∞))
104 f1ocnvfvb 7272 . . . . . . . . . . . 12 (((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))):(0[,)1)–1-1-onto→(0[,)+∞) ∧ 𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → (((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑥) = 𝑦 ↔ (◡(𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑦) = 𝑥))
10524, 104mp3an1 1450 . . . . . . . . . . 11 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → (((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑥) = 𝑦 ↔ (◡(𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑦) = 𝑥))
106 simpl 482 . . . . . . . . . . . . 13 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → 𝑥 ∈ (0[,)1))
107 ovex 7438 . . . . . . . . . . . . 13 (𝑥 / (1 − 𝑥)) ∈ V
10822fvmpt2 6997 . . . . . . . . . . . . 13 ((𝑥 ∈ (0[,)1) ∧ (𝑥 / (1 − 𝑥)) ∈ V) → ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑥) = (𝑥 / (1 − 𝑥)))
109106, 107, 108sylancl 586 . . . . . . . . . . . 12 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑥) = (𝑥 / (1 − 𝑥)))
110109eqeq1d 2737 . . . . . . . . . . 11 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → (((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑥) = 𝑦 ↔ (𝑥 / (1 − 𝑥)) = 𝑦))
111 simpr 484 . . . . . . . . . . . . 13 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → 𝑦 ∈ (0[,)+∞))
112 ovex 7438 . . . . . . . . . . . . 13 (𝑦 / (1 + 𝑦)) ∈ V
11351fvmpt2 6997 . . . . . . . . . . . . 13 ((𝑦 ∈ (0[,)+∞) ∧ (𝑦 / (1 + 𝑦)) ∈ V) → (◡(𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑦) = (𝑦 / (1 + 𝑦)))
114111, 112, 113sylancl 586 . . . . . . . . . . . 12 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → (◡(𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑦) = (𝑦 / (1 + 𝑦)))
115114eqeq1d 2737 . . . . . . . . . . 11 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → ((◡(𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑦) = 𝑥 ↔ (𝑦 / (1 + 𝑦)) = 𝑥))
116105, 110, 1153bitr3rd 310 . . . . . . . . . 10 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → ((𝑦 / (1 + 𝑦)) = 𝑥 ↔ (𝑥 / (1 − 𝑥)) = 𝑦))
117 eqcom 2742 . . . . . . . . . 10 (𝑥 = (𝑦 / (1 + 𝑦)) ↔ (𝑦 / (1 + 𝑦)) = 𝑥)
118 eqcom 2742 . . . . . . . . . 10 (𝑦 = (𝑥 / (1 − 𝑥)) ↔ (𝑥 / (1 − 𝑥)) = 𝑦)
119116, 117, 1183bitr4g 314 . . . . . . . . 9 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑦 = (𝑥 / (1 − 𝑥))))
12021, 47, 119syl2an 596 . . . . . . . 8 (((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 1) ∧ (𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞)) → (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑦 = (𝑥 / (1 − 𝑥))))
121120an4s 660 . . . . . . 7 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ (¬ 𝑥 = 1 ∧ ¬ 𝑦 = +∞)) → (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑦 = (𝑥 / (1 − 𝑥))))
122121anass1rs 655 . . . . . 6 ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑥 = 1) → (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑦 = (𝑥 / (1 − 𝑥))))
12386, 88, 103, 122ifbothda 4539 . . . . 5 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))))
12460, 62, 84, 123ifbothda 4539 . . . 4 ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) → (𝑥 = if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))) ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))))
125124adantl 481 . . 3 ((⊤ ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞))) → (𝑥 = if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))) ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))))
1261, 33, 58, 125f1ocnv2d 7660 . 2 (⊤ → (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))))))
127126mptru 1547 1 (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540  ⊤wtru 1541   ∈ wcel 2108   ≠ wne 2932   ∉ wnel 3036  ∀wral 3051  Vcvv 3459   ∪ cun 3924  ifcif 4500  {csn 4601   class class class wbr 5119   ↦ cmpt 5201  ◡ccnv 5653  ⟶wf 6527  –1-1-onto→wf1o 6530  ‘cfv 6531  (class class class)co 7405  ℝcr 11128  0cc0 11129  1c1 11130   + caddc 11132  +∞cpnf 11266  ℝ*cxr 11268   < clt 11269   ≤ cle 11270   − cmin 11466   / cdiv 11894  [,)cico 13364  [,]cicc 13365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-po 5561  df-so 5562  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-div 11895  df-rp 13009  df-ico 13368  df-icc 13369
This theorem is referenced by:  iccpnfhmeo  24894  xrhmeo  24895
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