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Theorem iccpnfcnv 24213
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 11128 . . . . . . 7 0 ∈ ℝ*
3 pnfxr 11135 . . . . . . 7 +∞ ∈ ℝ*
4 0lepnf 12974 . . . . . . 7 0 ≤ +∞
5 ubicc2 13303 . . . . . . 7 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞))
62, 3, 4, 5mp3an 1461 . . . . . 6 +∞ ∈ (0[,]+∞)
76a1i 11 . . . . 5 ((𝑥 ∈ (0[,]1) ∧ 𝑥 = 1) → +∞ ∈ (0[,]+∞))
8 icossicc 13274 . . . . . 6 (0[,)+∞) ⊆ (0[,]+∞)
9 1xr 11140 . . . . . . . . . . . . . 14 1 ∈ ℝ*
10 0le1 11604 . . . . . . . . . . . . . 14 0 ≤ 1
11 snunico 13317 . . . . . . . . . . . . . 14 ((0 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ 0 ≤ 1) → ((0[,)1) ∪ {1}) = (0[,]1))
122, 9, 10, 11mp3an 1461 . . . . . . . . . . . . 13 ((0[,)1) ∪ {1}) = (0[,]1)
1312eleq2i 2829 . . . . . . . . . . . 12 (𝑥 ∈ ((0[,)1) ∪ {1}) ↔ 𝑥 ∈ (0[,]1))
14 elun 4100 . . . . . . . . . . . 12 (𝑥 ∈ ((0[,)1) ∪ {1}) ↔ (𝑥 ∈ (0[,)1) ∨ 𝑥 ∈ {1}))
1513, 14bitr3i 277 . . . . . . . . . . 11 (𝑥 ∈ (0[,]1) ↔ (𝑥 ∈ (0[,)1) ∨ 𝑥 ∈ {1}))
16 pm2.53 849 . . . . . . . . . . 11 ((𝑥 ∈ (0[,)1) ∨ 𝑥 ∈ {1}) → (¬ 𝑥 ∈ (0[,)1) → 𝑥 ∈ {1}))
1715, 16sylbi 216 . . . . . . . . . 10 (𝑥 ∈ (0[,]1) → (¬ 𝑥 ∈ (0[,)1) → 𝑥 ∈ {1}))
18 elsni 4595 . . . . . . . . . 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 408 . . . . . . 7 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 1) → 𝑥 ∈ (0[,)1))
22 eqid 2737 . . . . . . . . . . . 12 (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) = (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))
2322icopnfcnv 24211 . . . . . . . . . . 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 485 . . . . . . . . . 10 (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))):(0[,)1)–1-1-onto→(0[,)+∞)
25 f1of 6772 . . . . . . . . . 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 7045 . . . . . . . . 9 (∀𝑥 ∈ (0[,)1)(𝑥 / (1 − 𝑥)) ∈ (0[,)+∞) ↔ (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))):(0[,)1)⟶(0[,)+∞))
2826, 27mpbir 230 . . . . . . . 8 ∀𝑥 ∈ (0[,)1)(𝑥 / (1 − 𝑥)) ∈ (0[,)+∞)
2928rspec 3230 . . . . . . 7 (𝑥 ∈ (0[,)1) → (𝑥 / (1 − 𝑥)) ∈ (0[,)+∞))
3021, 29syl 17 . . . . . 6 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 1) → (𝑥 / (1 − 𝑥)) ∈ (0[,)+∞))
318, 30sselid 3934 . . . . 5 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 1) → (𝑥 / (1 − 𝑥)) ∈ (0[,]+∞))
327, 31ifclda 4513 . . . 4 (𝑥 ∈ (0[,]1) → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ∈ (0[,]+∞))
3332adantl 483 . . 3 ((⊤ ∧ 𝑥 ∈ (0[,]1)) → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ∈ (0[,]+∞))
34 1elunit 13308 . . . . . 6 1 ∈ (0[,]1)
3534a1i 11 . . . . 5 ((𝑦 ∈ (0[,]+∞) ∧ 𝑦 = +∞) → 1 ∈ (0[,]1))
36 icossicc 13274 . . . . . 6 (0[,)1) ⊆ (0[,]1)
37 snunico 13317 . . . . . . . . . . . . . 14 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → ((0[,)+∞) ∪ {+∞}) = (0[,]+∞))
382, 3, 4, 37mp3an 1461 . . . . . . . . . . . . 13 ((0[,)+∞) ∪ {+∞}) = (0[,]+∞)
3938eleq2i 2829 . . . . . . . . . . . 12 (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ 𝑦 ∈ (0[,]+∞))
40 elun 4100 . . . . . . . . . . . 12 (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
4139, 40bitr3i 277 . . . . . . . . . . 11 (𝑦 ∈ (0[,]+∞) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
42 pm2.53 849 . . . . . . . . . . 11 ((𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
4341, 42sylbi 216 . . . . . . . . . 10 (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
44 elsni 4595 . . . . . . . . . 10 (𝑦 ∈ {+∞} → 𝑦 = +∞)
4543, 44syl6 35 . . . . . . . . 9 (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 = +∞))
4645con1d 145 . . . . . . . 8 (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 = +∞ → 𝑦 ∈ (0[,)+∞)))
4746imp 408 . . . . . . 7 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 𝑦 ∈ (0[,)+∞))
48 f1ocnv 6784 . . . . . . . . . 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 6772 . . . . . . . . . 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 487 . . . . . . . . . 10 ◡(𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) = (𝑦 ∈ (0[,)+∞) ↦ (𝑦 / (1 + 𝑦)))
5251fmpt 7045 . . . . . . . . 9 (∀𝑦 ∈ (0[,)+∞)(𝑦 / (1 + 𝑦)) ∈ (0[,)1) ↔ ◡(𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))):(0[,)+∞)⟶(0[,)1))
5350, 52mpbir 230 . . . . . . . 8 ∀𝑦 ∈ (0[,)+∞)(𝑦 / (1 + 𝑦)) ∈ (0[,)1)
5453rspec 3230 . . . . . . 7 (𝑦 ∈ (0[,)+∞) → (𝑦 / (1 + 𝑦)) ∈ (0[,)1))
5547, 54syl 17 . . . . . 6 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (𝑦 / (1 + 𝑦)) ∈ (0[,)1))
5636, 55sselid 3934 . . . . 5 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (𝑦 / (1 + 𝑦)) ∈ (0[,]1))
5735, 56ifclda 4513 . . . 4 (𝑦 ∈ (0[,]+∞) → if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))) ∈ (0[,]1))
5857adantl 483 . . 3 ((⊤ ∧ 𝑦 ∈ (0[,]+∞)) → if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))) ∈ (0[,]1))
59 eqeq2 2749 . . . . . 6 (1 = if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))) → (𝑥 = 1 ↔ 𝑥 = if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦)))))
6059bibi1d 344 . . . . 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 2749 . . . . . 6 ((𝑦 / (1 + 𝑦)) = if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))) → (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑥 = if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦)))))
6261bibi1d 344 . . . . 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 486 . . . . . . 7 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → 𝑦 = +∞)
64 iftrue 4484 . . . . . . . 8 (𝑥 = 1 → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) = +∞)
6564eqeq2d 2748 . . . . . . 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 11122 . . . . . . . . 9 +∞ ∉ ℝ
68 neleq1 3052 . . . . . . . . . 10 (𝑦 = +∞ → (𝑦 ∉ ℝ ↔ +∞ ∉ ℝ))
6968adantl 483 . . . . . . . . 9 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑦 ∉ ℝ ↔ +∞ ∉ ℝ))
7067, 69mpbiri 258 . . . . . . . 8 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → 𝑦 ∉ ℝ)
71 neleq1 3052 . . . . . . . 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 3048 . . . . . . . 8 (if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ∉ ℝ ↔ ¬ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ∈ ℝ)
74 iffalse 4487 . . . . . . . . . . . . 13 (¬ 𝑥 = 1 → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) = (𝑥 / (1 − 𝑥)))
7574adantl 483 . . . . . . . . . . . 12 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 1) → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) = (𝑥 / (1 − 𝑥)))
76 rge0ssre 13294 . . . . . . . . . . . . 13 (0[,)+∞) ⊆ ℝ
7776, 30sselid 3934 . . . . . . . . . . . 12 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 1) → (𝑥 / (1 − 𝑥)) ∈ ℝ)
7875, 77eqeltrd 2838 . . . . . . . . . . 11 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 1) → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ∈ ℝ)
7978ex 414 . . . . . . . . . 10 (𝑥 ∈ (0[,]1) → (¬ 𝑥 = 1 → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ∈ ℝ))
8079ad2antrr 724 . . . . . . . . 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 241 . . . . . . 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 211 . . . . 5 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑥 = 1 ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))))
85 eqeq2 2749 . . . . . . 7 (+∞ = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) → (𝑦 = +∞ ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))))
8685bibi2d 343 . . . . . 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 2749 . . . . . . 7 ((𝑥 / (1 − 𝑥)) = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) → (𝑦 = (𝑥 / (1 − 𝑥)) ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))))
8887bibi2d 343 . . . . . 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 11083 . . . . . . . . . . . . . . 15 0 ∈ ℝ
90 elico2 13249 . . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . . 14 ((𝑦 / (1 + 𝑦)) ∈ (0[,)1) ↔ ((𝑦 / (1 + 𝑦)) ∈ ℝ ∧ 0 ≤ (𝑦 / (1 + 𝑦)) ∧ (𝑦 / (1 + 𝑦)) < 1))
9255, 91sylib 217 . . . . . . . . . . . . 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 11216 . . . . . . . . . . 11 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 1 ≠ (𝑦 / (1 + 𝑦)))
9695adantll 712 . . . . . . . . . 10 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → 1 ≠ (𝑦 / (1 + 𝑦)))
9796neneqd 2946 . . . . . . . . 9 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → ¬ 1 = (𝑦 / (1 + 𝑦)))
98 eqeq1 2741 . . . . . . . . . 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 408 . . . . . . 7 ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 1) → ¬ 𝑥 = (𝑦 / (1 + 𝑦)))
102 simplr 767 . . . . . . 7 ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 1) → ¬ 𝑦 = +∞)
103101, 1022falsed 377 . . . . . 6 ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 1) → (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑦 = +∞))
104 f1ocnvfvb 7212 . . . . . . . . . . . 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 1448 . . . . . . . . . . 11 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → (((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑥) = 𝑦 ↔ (◡(𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑦) = 𝑥))
106 simpl 484 . . . . . . . . . . . . 13 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → 𝑥 ∈ (0[,)1))
107 ovex 7375 . . . . . . . . . . . . 13 (𝑥 / (1 − 𝑥)) ∈ V
10822fvmpt2 6947 . . . . . . . . . . . . 13 ((𝑥 ∈ (0[,)1) ∧ (𝑥 / (1 − 𝑥)) ∈ V) → ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑥) = (𝑥 / (1 − 𝑥)))
109106, 107, 108sylancl 587 . . . . . . . . . . . 12 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑥) = (𝑥 / (1 − 𝑥)))
110109eqeq1d 2739 . . . . . . . . . . 11 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → (((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑥) = 𝑦 ↔ (𝑥 / (1 − 𝑥)) = 𝑦))
111 simpr 486 . . . . . . . . . . . . 13 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → 𝑦 ∈ (0[,)+∞))
112 ovex 7375 . . . . . . . . . . . . 13 (𝑦 / (1 + 𝑦)) ∈ V
11351fvmpt2 6947 . . . . . . . . . . . . 13 ((𝑦 ∈ (0[,)+∞) ∧ (𝑦 / (1 + 𝑦)) ∈ V) → (◡(𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑦) = (𝑦 / (1 + 𝑦)))
114111, 112, 113sylancl 587 . . . . . . . . . . . 12 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → (◡(𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑦) = (𝑦 / (1 + 𝑦)))
115114eqeq1d 2739 . . . . . . . . . . 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 2744 . . . . . . . . . 10 (𝑥 = (𝑦 / (1 + 𝑦)) ↔ (𝑦 / (1 + 𝑦)) = 𝑥)
118 eqcom 2744 . . . . . . . . . 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 597 . . . . . . . 8 (((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 1) ∧ (𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞)) → (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑦 = (𝑥 / (1 − 𝑥))))
121120an4s 658 . . . . . . 7 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ (¬ 𝑥 = 1 ∧ ¬ 𝑦 = +∞)) → (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑦 = (𝑥 / (1 − 𝑥))))
122121anass1rs 653 . . . . . 6 ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑥 = 1) → (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑦 = (𝑥 / (1 − 𝑥))))
12386, 88, 103, 122ifbothda 4516 . . . . 5 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))))
12460, 62, 84, 123ifbothda 4516 . . . 4 ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) → (𝑥 = if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))) ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))))
125124adantl 483 . . 3 ((⊤ ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞))) → (𝑥 = if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))) ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))))
1261, 33, 58, 125f1ocnv2d 7589 . 2 (⊤ → (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))))))
127126mptru 1548 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 205   ∧ wa 397   ∨ wo 845   ∧ w3a 1087   = wceq 1541  ⊤wtru 1542   ∈ wcel 2106   ≠ wne 2941   ∉ wnel 3047  ∀wral 3062  Vcvv 3442   ∪ cun 3900  ifcif 4478  {csn 4578   class class class wbr 5097   ↦ cmpt 5180  ◡ccnv 5624  ⟶wf 6480  –1-1-onto→wf1o 6483  ‘cfv 6484  (class class class)co 7342  ℝcr 10976  0cc0 10977  1c1 10978   + caddc 10980  +∞cpnf 11112  ℝ*cxr 11114   < clt 11115   ≤ cle 11116   − cmin 11311   / cdiv 11738  [,)cico 13187  [,]cicc 13188
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 2708  ax-sep 5248  ax-nul 5255  ax-pow 5313  ax-pr 5377  ax-un 7655  ax-cnex 11033  ax-resscn 11034  ax-1cn 11035  ax-icn 11036  ax-addcl 11037  ax-addrcl 11038  ax-mulcl 11039  ax-mulrcl 11040  ax-mulcom 11041  ax-addass 11042  ax-mulass 11043  ax-distr 11044  ax-i2m1 11045  ax-1ne0 11046  ax-1rid 11047  ax-rnegex 11048  ax-rrecex 11049  ax-cnre 11050  ax-pre-lttri 11051  ax-pre-lttrn 11052  ax-pre-ltadd 11053  ax-pre-mulgt0 11054
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3732  df-csb 3848  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4275  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-br 5098  df-opab 5160  df-mpt 5181  df-id 5523  df-po 5537  df-so 5538  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-f1 6489  df-fo 6490  df-f1o 6491  df-fv 6492  df-riota 7298  df-ov 7345  df-oprab 7346  df-mpo 7347  df-er 8574  df-en 8810  df-dom 8811  df-sdom 8812  df-pnf 11117  df-mnf 11118  df-xr 11119  df-ltxr 11120  df-le 11121  df-sub 11313  df-neg 11314  df-div 11739  df-rp 12837  df-ico 13191  df-icc 13192
This theorem is referenced by:  iccpnfhmeo  24214  xrhmeo  24215
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