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Theorem iccpnfcnv 23795
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 10845 . . . . . . 7 0 ∈ ℝ*
3 pnfxr 10852 . . . . . . 7 +∞ ∈ ℝ*
4 0lepnf 12689 . . . . . . 7 0 ≤ +∞
5 ubicc2 13018 . . . . . . 7 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞))
62, 3, 4, 5mp3an 1463 . . . . . 6 +∞ ∈ (0[,]+∞)
76a1i 11 . . . . 5 ((𝑥 ∈ (0[,]1) ∧ 𝑥 = 1) → +∞ ∈ (0[,]+∞))
8 icossicc 12989 . . . . . 6 (0[,)+∞) ⊆ (0[,]+∞)
9 1xr 10857 . . . . . . . . . . . . . 14 1 ∈ ℝ*
10 0le1 11320 . . . . . . . . . . . . . 14 0 ≤ 1
11 snunico 13032 . . . . . . . . . . . . . 14 ((0 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ 0 ≤ 1) → ((0[,)1) ∪ {1}) = (0[,]1))
122, 9, 10, 11mp3an 1463 . . . . . . . . . . . . 13 ((0[,)1) ∪ {1}) = (0[,]1)
1312eleq2i 2822 . . . . . . . . . . . 12 (𝑥 ∈ ((0[,)1) ∪ {1}) ↔ 𝑥 ∈ (0[,]1))
14 elun 4049 . . . . . . . . . . . 12 (𝑥 ∈ ((0[,)1) ∪ {1}) ↔ (𝑥 ∈ (0[,)1) ∨ 𝑥 ∈ {1}))
1513, 14bitr3i 280 . . . . . . . . . . 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 220 . . . . . . . . . 10 (𝑥 ∈ (0[,]1) → (¬ 𝑥 ∈ (0[,)1) → 𝑥 ∈ {1}))
18 elsni 4544 . . . . . . . . . 10 (𝑥 ∈ {1} → 𝑥 = 1)
1917, 18syl6 35 . . . . . . . . 9 (𝑥 ∈ (0[,]1) → (¬ 𝑥 ∈ (0[,)1) → 𝑥 = 1))
2019con1d 147 . . . . . . . 8 (𝑥 ∈ (0[,]1) → (¬ 𝑥 = 1 → 𝑥 ∈ (0[,)1)))
2120imp 410 . . . . . . 7 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 1) → 𝑥 ∈ (0[,)1))
22 eqid 2736 . . . . . . . . . . . 12 (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) = (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))
2322icopnfcnv 23793 . . . . . . . . . . 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 487 . . . . . . . . . 10 (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))):(0[,)1)–1-1-onto→(0[,)+∞)
25 f1of 6639 . . . . . . . . . 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 6905 . . . . . . . . 9 (∀𝑥 ∈ (0[,)1)(𝑥 / (1 − 𝑥)) ∈ (0[,)+∞) ↔ (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))):(0[,)1)⟶(0[,)+∞))
2826, 27mpbir 234 . . . . . . . 8 ∀𝑥 ∈ (0[,)1)(𝑥 / (1 − 𝑥)) ∈ (0[,)+∞)
2928rspec 3119 . . . . . . 7 (𝑥 ∈ (0[,)1) → (𝑥 / (1 − 𝑥)) ∈ (0[,)+∞))
3021, 29syl 17 . . . . . 6 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 1) → (𝑥 / (1 − 𝑥)) ∈ (0[,)+∞))
318, 30sseldi 3885 . . . . 5 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 1) → (𝑥 / (1 − 𝑥)) ∈ (0[,]+∞))
327, 31ifclda 4460 . . . 4 (𝑥 ∈ (0[,]1) → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ∈ (0[,]+∞))
3332adantl 485 . . 3 ((⊤ ∧ 𝑥 ∈ (0[,]1)) → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ∈ (0[,]+∞))
34 1elunit 13023 . . . . . 6 1 ∈ (0[,]1)
3534a1i 11 . . . . 5 ((𝑦 ∈ (0[,]+∞) ∧ 𝑦 = +∞) → 1 ∈ (0[,]1))
36 icossicc 12989 . . . . . 6 (0[,)1) ⊆ (0[,]1)
37 snunico 13032 . . . . . . . . . . . . . 14 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → ((0[,)+∞) ∪ {+∞}) = (0[,]+∞))
382, 3, 4, 37mp3an 1463 . . . . . . . . . . . . 13 ((0[,)+∞) ∪ {+∞}) = (0[,]+∞)
3938eleq2i 2822 . . . . . . . . . . . 12 (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ 𝑦 ∈ (0[,]+∞))
40 elun 4049 . . . . . . . . . . . 12 (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
4139, 40bitr3i 280 . . . . . . . . . . 11 (𝑦 ∈ (0[,]+∞) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
42 pm2.53 851 . . . . . . . . . . 11 ((𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
4341, 42sylbi 220 . . . . . . . . . 10 (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
44 elsni 4544 . . . . . . . . . 10 (𝑦 ∈ {+∞} → 𝑦 = +∞)
4543, 44syl6 35 . . . . . . . . 9 (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 = +∞))
4645con1d 147 . . . . . . . 8 (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 = +∞ → 𝑦 ∈ (0[,)+∞)))
4746imp 410 . . . . . . 7 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 𝑦 ∈ (0[,)+∞))
48 f1ocnv 6651 . . . . . . . . . 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 6639 . . . . . . . . . 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 489 . . . . . . . . . 10 ◡(𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) = (𝑦 ∈ (0[,)+∞) ↦ (𝑦 / (1 + 𝑦)))
5251fmpt 6905 . . . . . . . . 9 (∀𝑦 ∈ (0[,)+∞)(𝑦 / (1 + 𝑦)) ∈ (0[,)1) ↔ ◡(𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))):(0[,)+∞)⟶(0[,)1))
5350, 52mpbir 234 . . . . . . . 8 ∀𝑦 ∈ (0[,)+∞)(𝑦 / (1 + 𝑦)) ∈ (0[,)1)
5453rspec 3119 . . . . . . 7 (𝑦 ∈ (0[,)+∞) → (𝑦 / (1 + 𝑦)) ∈ (0[,)1))
5547, 54syl 17 . . . . . 6 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (𝑦 / (1 + 𝑦)) ∈ (0[,)1))
5636, 55sseldi 3885 . . . . 5 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (𝑦 / (1 + 𝑦)) ∈ (0[,]1))
5735, 56ifclda 4460 . . . 4 (𝑦 ∈ (0[,]+∞) → if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))) ∈ (0[,]1))
5857adantl 485 . . 3 ((⊤ ∧ 𝑦 ∈ (0[,]+∞)) → if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))) ∈ (0[,]1))
59 eqeq2 2748 . . . . . 6 (1 = if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))) → (𝑥 = 1 ↔ 𝑥 = if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦)))))
6059bibi1d 347 . . . . 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 2748 . . . . . 6 ((𝑦 / (1 + 𝑦)) = if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))) → (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑥 = if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦)))))
6261bibi1d 347 . . . . 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 488 . . . . . . 7 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → 𝑦 = +∞)
64 iftrue 4431 . . . . . . . 8 (𝑥 = 1 → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) = +∞)
6564eqeq2d 2747 . . . . . . 7 (𝑥 = 1 → (𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ↔ 𝑦 = +∞))
6663, 65syl5ibrcom 250 . . . . . 6 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑥 = 1 → 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))))
67 pnfnre 10839 . . . . . . . . 9 +∞ ∉ ℝ
68 neleq1 3041 . . . . . . . . . 10 (𝑦 = +∞ → (𝑦 ∉ ℝ ↔ +∞ ∉ ℝ))
6968adantl 485 . . . . . . . . 9 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑦 ∉ ℝ ↔ +∞ ∉ ℝ))
7067, 69mpbiri 261 . . . . . . . 8 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → 𝑦 ∉ ℝ)
71 neleq1 3041 . . . . . . . 8 (𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) → (𝑦 ∉ ℝ ↔ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ∉ ℝ))
7270, 71syl5ibcom 248 . . . . . . 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 4434 . . . . . . . . . . . . 13 (¬ 𝑥 = 1 → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) = (𝑥 / (1 − 𝑥)))
7574adantl 485 . . . . . . . . . . . 12 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 1) → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) = (𝑥 / (1 − 𝑥)))
76 rge0ssre 13009 . . . . . . . . . . . . 13 (0[,)+∞) ⊆ ℝ
7776, 30sseldi 3885 . . . . . . . . . . . 12 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 1) → (𝑥 / (1 − 𝑥)) ∈ ℝ)
7875, 77eqeltrd 2831 . . . . . . . . . . 11 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 1) → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ∈ ℝ)
7978ex 416 . . . . . . . . . 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 147 . . . . . . . 8 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (¬ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) ∈ ℝ → 𝑥 = 1))
8273, 81syl5bi 245 . . . . . . 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 215 . . . . 5 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑥 = 1 ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))))
85 eqeq2 2748 . . . . . . 7 (+∞ = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) → (𝑦 = +∞ ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))))
8685bibi2d 346 . . . . . 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 2748 . . . . . . 7 ((𝑥 / (1 − 𝑥)) = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) → (𝑦 = (𝑥 / (1 − 𝑥)) ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))))
8887bibi2d 346 . . . . . 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 10800 . . . . . . . . . . . . . . 15 0 ∈ ℝ
90 elico2 12964 . . . . . . . . . . . . . . 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 221 . . . . . . . . . . . . 13 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → ((𝑦 / (1 + 𝑦)) ∈ ℝ ∧ 0 ≤ (𝑦 / (1 + 𝑦)) ∧ (𝑦 / (1 + 𝑦)) < 1))
9392simp1d 1144 . . . . . . . . . . . 12 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (𝑦 / (1 + 𝑦)) ∈ ℝ)
9492simp3d 1146 . . . . . . . . . . . 12 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (𝑦 / (1 + 𝑦)) < 1)
9593, 94gtned 10932 . . . . . . . . . . 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 2740 . . . . . . . . . 10 (𝑥 = 1 → (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 1 = (𝑦 / (1 + 𝑦))))
9998notbid 321 . . . . . . . . 9 (𝑥 = 1 → (¬ 𝑥 = (𝑦 / (1 + 𝑦)) ↔ ¬ 1 = (𝑦 / (1 + 𝑦))))
10097, 99syl5ibrcom 250 . . . . . . . 8 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → (𝑥 = 1 → ¬ 𝑥 = (𝑦 / (1 + 𝑦))))
101100imp 410 . . . . . . 7 ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 1) → ¬ 𝑥 = (𝑦 / (1 + 𝑦)))
102 simplr 769 . . . . . . 7 ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 1) → ¬ 𝑦 = +∞)
103101, 1022falsed 380 . . . . . 6 ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 1) → (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑦 = +∞))
104 f1ocnvfvb 7068 . . . . . . . . . . . 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 486 . . . . . . . . . . . . 13 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → 𝑥 ∈ (0[,)1))
107 ovex 7224 . . . . . . . . . . . . 13 (𝑥 / (1 − 𝑥)) ∈ V
10822fvmpt2 6807 . . . . . . . . . . . . 13 ((𝑥 ∈ (0[,)1) ∧ (𝑥 / (1 − 𝑥)) ∈ V) → ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑥) = (𝑥 / (1 − 𝑥)))
109106, 107, 108sylancl 589 . . . . . . . . . . . 12 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑥) = (𝑥 / (1 − 𝑥)))
110109eqeq1d 2738 . . . . . . . . . . 11 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → (((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑥) = 𝑦 ↔ (𝑥 / (1 − 𝑥)) = 𝑦))
111 simpr 488 . . . . . . . . . . . . 13 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → 𝑦 ∈ (0[,)+∞))
112 ovex 7224 . . . . . . . . . . . . 13 (𝑦 / (1 + 𝑦)) ∈ V
11351fvmpt2 6807 . . . . . . . . . . . . 13 ((𝑦 ∈ (0[,)+∞) ∧ (𝑦 / (1 + 𝑦)) ∈ V) → (◡(𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑦) = (𝑦 / (1 + 𝑦)))
114111, 112, 113sylancl 589 . . . . . . . . . . . 12 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → (◡(𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑦) = (𝑦 / (1 + 𝑦)))
115114eqeq1d 2738 . . . . . . . . . . 11 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → ((◡(𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑦) = 𝑥 ↔ (𝑦 / (1 + 𝑦)) = 𝑥))
116105, 110, 1153bitr3rd 313 . . . . . . . . . 10 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → ((𝑦 / (1 + 𝑦)) = 𝑥 ↔ (𝑥 / (1 − 𝑥)) = 𝑦))
117 eqcom 2743 . . . . . . . . . 10 (𝑥 = (𝑦 / (1 + 𝑦)) ↔ (𝑦 / (1 + 𝑦)) = 𝑥)
118 eqcom 2743 . . . . . . . . . 10 (𝑦 = (𝑥 / (1 − 𝑥)) ↔ (𝑥 / (1 − 𝑥)) = 𝑦)
119116, 117, 1183bitr4g 317 . . . . . . . . 9 ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑦 = (𝑥 / (1 − 𝑥))))
12021, 47, 119syl2an 599 . . . . . . . 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 4463 . . . . 5 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → (𝑥 = (𝑦 / (1 + 𝑦)) ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))))
12460, 62, 84, 123ifbothda 4463 . . . 4 ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) → (𝑥 = if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))) ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))))
125124adantl 485 . . 3 ((⊤ ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞))) → (𝑥 = if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))) ↔ 𝑦 = if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))))
1261, 33, 58, 125f1ocnv2d 7436 . 2 (⊤ → (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))))))
127126mptru 1550 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 209   ∧ wa 399   ∨ wo 847   ∧ w3a 1089   = wceq 1543  ⊤wtru 1544   ∈ wcel 2112   ≠ wne 2932   ∉ wnel 3036  ∀wral 3051  Vcvv 3398   ∪ cun 3851  ifcif 4425  {csn 4527   class class class wbr 5039   ↦ cmpt 5120  ◡ccnv 5535  ⟶wf 6354  –1-1-onto→wf1o 6357  ‘cfv 6358  (class class class)co 7191  ℝcr 10693  0cc0 10694  1c1 10695   + caddc 10697  +∞cpnf 10829  ℝ*cxr 10831   < clt 10832   ≤ cle 10833   − cmin 11027   / cdiv 11454  [,)cico 12902  [,]cicc 12903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501  ax-cnex 10750  ax-resscn 10751  ax-1cn 10752  ax-icn 10753  ax-addcl 10754  ax-addrcl 10755  ax-mulcl 10756  ax-mulrcl 10757  ax-mulcom 10758  ax-addass 10759  ax-mulass 10760  ax-distr 10761  ax-i2m1 10762  ax-1ne0 10763  ax-1rid 10764  ax-rnegex 10765  ax-rrecex 10766  ax-cnre 10767  ax-pre-lttri 10768  ax-pre-lttrn 10769  ax-pre-ltadd 10770  ax-pre-mulgt0 10771
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-po 5453  df-so 5454  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7148  df-ov 7194  df-oprab 7195  df-mpo 7196  df-er 8369  df-en 8605  df-dom 8606  df-sdom 8607  df-pnf 10834  df-mnf 10835  df-xr 10836  df-ltxr 10837  df-le 10838  df-sub 11029  df-neg 11030  df-div 11455  df-rp 12552  df-ico 12906  df-icc 12907
This theorem is referenced by:  iccpnfhmeo  23796  xrhmeo  23797
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