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Theorem iccpnfcnv 24994
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 11337 . . . . . . 7 0 ∈ ℝ*
3 pnfxr 11344 . . . . . . 7 +∞ ∈ ℝ*
4 0lepnf 13195 . . . . . . 7 0 ≤ +∞
5 ubicc2 13525 . . . . . . 7 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞))
62, 3, 4, 5mp3an 1461 . . . . . 6 +∞ ∈ (0[,]+∞)
76a1i 11 . . . . 5 ((𝑥 ∈ (0[,]1) ∧ 𝑥 = 1) → +∞ ∈ (0[,]+∞))
8 icossicc 13496 . . . . . 6 (0[,)+∞) ⊆ (0[,]+∞)
9 1xr 11349 . . . . . . . . . . . . . 14 1 ∈ ℝ*
10 0le1 11813 . . . . . . . . . . . . . 14 0 ≤ 1
11 snunico 13539 . . . . . . . . . . . . . 14 ((0 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ 0 ≤ 1) → ((0[,)1) ∪ {1}) = (0[,]1))
122, 9, 10, 11mp3an 1461 . . . . . . . . . . . . 13 ((0[,)1) ∪ {1}) = (0[,]1)
1312eleq2i 2836 . . . . . . . . . . . 12 (𝑥 ∈ ((0[,)1) ∪ {1}) ↔ 𝑥 ∈ (0[,]1))
14 elun 4176 . . . . . . . . . . . 12 (𝑥 ∈ ((0[,)1) ∪ {1}) ↔ (𝑥 ∈ (0[,)1) ∨ 𝑥 ∈ {1}))
1513, 14bitr3i 277 . . . . . . . . . . 11 (𝑥 ∈ (0[,]1) ↔ (𝑥 ∈ (0[,)1) ∨ 𝑥 ∈ {1}))
16 pm2.53 850 . . . . . . . . . . 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 4665 . . . . . . . . . 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 2740 . . . . . . . . . . . 12 (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) = (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))
2322icopnfcnv 24992 . . . . . . . . . . 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 6862 . . . . . . . . . 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 7144 . . . . . . . . 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 3256 . . . . . . 7 (𝑥 ∈ (0[,)1) → (𝑥 / (1 − 𝑥)) ∈ (0[,)+∞))
3021, 29syl 17 . . . . . 6 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 1) → (𝑥 / (1 − 𝑥)) ∈ (0[,)+∞))
318, 30sselid 4006 . . . . 5 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 1) → (𝑥 / (1 − 𝑥)) ∈ (0[,]+∞))
327, 31ifclda 4583 . . . 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 13530 . . . . . 6 1 ∈ (0[,]1)
3534a1i 11 . . . . 5 ((𝑦 ∈ (0[,]+∞) ∧ 𝑦 = +∞) → 1 ∈ (0[,]1))
36 icossicc 13496 . . . . . 6 (0[,)1) ⊆ (0[,]1)
37 snunico 13539 . . . . . . . . . . . . . 14 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → ((0[,)+∞) ∪ {+∞}) = (0[,]+∞))
382, 3, 4, 37mp3an 1461 . . . . . . . . . . . . 13 ((0[,)+∞) ∪ {+∞}) = (0[,]+∞)
3938eleq2i 2836 . . . . . . . . . . . 12 (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ 𝑦 ∈ (0[,]+∞))
40 elun 4176 . . . . . . . . . . . 12 (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
4139, 40bitr3i 277 . . . . . . . . . . 11 (𝑦 ∈ (0[,]+∞) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
42 pm2.53 850 . . . . . . . . . . 11 ((𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
4341, 42sylbi 217 . . . . . . . . . 10 (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
44 elsni 4665 . . . . . . . . . 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 6874 . . . . . . . . . 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 6862 . . . . . . . . . 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 7144 . . . . . . . . 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 3256 . . . . . . 7 (𝑦 ∈ (0[,)+∞) → (𝑦 / (1 + 𝑦)) ∈ (0[,)1))
5547, 54syl 17 . . . . . 6 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (𝑦 / (1 + 𝑦)) ∈ (0[,)1))
5636, 55sselid 4006 . . . . 5 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (𝑦 / (1 + 𝑦)) ∈ (0[,]1))
5735, 56ifclda 4583 . . . 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 2752 . . . . . 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 2752 . . . . . 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 4554 . . . . . . . 8 (𝑥 = 1 → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) = +∞)
6564eqeq2d 2751 . . . . . . 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 11331 . . . . . . . . 9 +∞ ∉ ℝ
68 neleq1 3058 . . . . . . . . . 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 3058 . . . . . . . 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 3053 . . . . . . . 8 (if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ∉ ℝ ↔ ¬ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ∈ ℝ)
74 iffalse 4557 . . . . . . . . . . . . 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 13516 . . . . . . . . . . . . 13 (0[,)+∞) ⊆ ℝ
7776, 30sselid 4006 . . . . . . . . . . . 12 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 1) → (𝑥 / (1 − 𝑥)) ∈ ℝ)
7875, 77eqeltrd 2844 . . . . . . . . . . 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 725 . . . . . . . . 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 2752 . . . . . . 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 2752 . . . . . . 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 11292 . . . . . . . . . . . . . . 15 0 ∈ ℝ
90 elico2 13471 . . . . . . . . . . . . . . 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 691 . . . . . . . . . . . . . 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 11425 . . . . . . . . . . 11 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 1 ≠ (𝑦 / (1 + 𝑦)))
9695adantll 713 . . . . . . . . . 10 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → 1 ≠ (𝑦 / (1 + 𝑦)))
9796neneqd 2951 . . . . . . . . 9 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → ¬ 1 = (𝑦 / (1 + 𝑦)))
98 eqeq1 2744 . . . . . . . . . 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 7315 . . . . . . . . . . . 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 482 . . . . . . . . . . . . 13 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → 𝑥 ∈ (0[,)1))
107 ovex 7481 . . . . . . . . . . . . 13 (𝑥 / (1 − 𝑥)) ∈ V
10822fvmpt2 7040 . . . . . . . . . . . . 13 ((𝑥 ∈ (0[,)1) ∧ (𝑥 / (1 − 𝑥)) ∈ V) → ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑥) = (𝑥 / (1 − 𝑥)))
109106, 107, 108sylancl 585 . . . . . . . . . . . 12 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑥) = (𝑥 / (1 − 𝑥)))
110109eqeq1d 2742 . . . . . . . . . . 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 7481 . . . . . . . . . . . . 13 (𝑦 / (1 + 𝑦)) ∈ V
11351fvmpt2 7040 . . . . . . . . . . . . 13 ((𝑦 ∈ (0[,)+∞) ∧ (𝑦 / (1 + 𝑦)) ∈ V) → (◡(𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑦) = (𝑦 / (1 + 𝑦)))
114111, 112, 113sylancl 585 . . . . . . . . . . . 12 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → (◡(𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑦) = (𝑦 / (1 + 𝑦)))
115114eqeq1d 2742 . . . . . . . . . . 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 2747 . . . . . . . . . 10 (𝑥 = (𝑦 / (1 + 𝑦)) ↔ (𝑦 / (1 + 𝑦)) = 𝑥)
118 eqcom 2747 . . . . . . . . . 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 595 . . . . . . . 8 (((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 1) ∧ (𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞)) → (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑦 = (𝑥 / (1 − 𝑥))))
121120an4s 659 . . . . . . 7 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ (¬ 𝑥 = 1 ∧ ¬ 𝑦 = +∞)) → (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑦 = (𝑥 / (1 − 𝑥))))
122121anass1rs 654 . . . . . 6 ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑥 = 1) → (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑦 = (𝑥 / (1 − 𝑥))))
12386, 88, 103, 122ifbothda 4586 . . . . 5 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))))
12460, 62, 84, 123ifbothda 4586 . . . 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 7703 . 2 (⊤ → (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))))))
127126mptru 1544 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 846   ∧ w3a 1087   = wceq 1537  ⊤wtru 1538   ∈ wcel 2108   ≠ wne 2946   ∉ wnel 3052  ∀wral 3067  Vcvv 3488   ∪ cun 3974  ifcif 4548  {csn 4648   class class class wbr 5166   ↦ cmpt 5249  ◡ccnv 5699  ⟶wf 6569  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448  ℝcr 11183  0cc0 11184  1c1 11185   + caddc 11187  +∞cpnf 11321  ℝ*cxr 11323   < clt 11324   ≤ cle 11325   − cmin 11520   / cdiv 11947  [,)cico 13409  [,]cicc 13410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-po 5607  df-so 5608  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-rp 13058  df-ico 13413  df-icc 13414
This theorem is referenced by:  iccpnfhmeo  24995  xrhmeo  24996
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