Users' Mathboxes Mathbox for ML < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fvineqsnf1 Structured version   Visualization version   GIF version

Theorem fvineqsnf1 37325
Description: A theorem about functions where the image of every point intersects the domain only at that point. If 𝐽 is a topology and 𝐴 is a set with no limit points, then there exists an 𝐹 such that this antecedent is true. See nlpfvineqsn 37324 for a proof of this fact. (Contributed by ML, 23-Mar-2021.)
Assertion
Ref Expression
fvineqsnf1 ((𝐹:𝐴𝐽 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → 𝐹:𝐴1-1𝐽)
Distinct variable groups:   𝐴,𝑝   𝐹,𝑝
Allowed substitution hint:   𝐽(𝑝)

Proof of Theorem fvineqsnf1
Dummy variable 𝑞 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6919 . . . . . . . . . 10 (𝑝 = 𝑞 → (𝐹𝑝) = (𝐹𝑞))
21ineq1d 4234 . . . . . . . . 9 (𝑝 = 𝑞 → ((𝐹𝑝) ∩ 𝐴) = ((𝐹𝑞) ∩ 𝐴))
3 sneq 4658 . . . . . . . . 9 (𝑝 = 𝑞 → {𝑝} = {𝑞})
42, 3eqeq12d 2750 . . . . . . . 8 (𝑝 = 𝑞 → (((𝐹𝑝) ∩ 𝐴) = {𝑝} ↔ ((𝐹𝑞) ∩ 𝐴) = {𝑞}))
54cbvralvw 3238 . . . . . . 7 (∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} ↔ ∀𝑞𝐴 ((𝐹𝑞) ∩ 𝐴) = {𝑞})
65biimpi 216 . . . . . 6 (∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} → ∀𝑞𝐴 ((𝐹𝑞) ∩ 𝐴) = {𝑞})
7 ax-5 1909 . . . . . . . 8 (∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} → ∀𝑞𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝})
8 alral 3077 . . . . . . . 8 (∀𝑞𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} → ∀𝑞𝐴𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝})
9 ralcom 3290 . . . . . . . . 9 (∀𝑞𝐴𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} ↔ ∀𝑝𝐴𝑞𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝})
109biimpi 216 . . . . . . . 8 (∀𝑞𝐴𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} → ∀𝑝𝐴𝑞𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝})
117, 8, 103syl 18 . . . . . . 7 (∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} → ∀𝑝𝐴𝑞𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝})
12 ax-5 1909 . . . . . . . 8 (∀𝑞𝐴 ((𝐹𝑞) ∩ 𝐴) = {𝑞} → ∀𝑝𝑞𝐴 ((𝐹𝑞) ∩ 𝐴) = {𝑞})
13 alral 3077 . . . . . . . 8 (∀𝑝𝑞𝐴 ((𝐹𝑞) ∩ 𝐴) = {𝑞} → ∀𝑝𝐴𝑞𝐴 ((𝐹𝑞) ∩ 𝐴) = {𝑞})
1412, 13syl 17 . . . . . . 7 (∀𝑞𝐴 ((𝐹𝑞) ∩ 𝐴) = {𝑞} → ∀𝑝𝐴𝑞𝐴 ((𝐹𝑞) ∩ 𝐴) = {𝑞})
1511, 14anim12i 612 . . . . . 6 ((∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} ∧ ∀𝑞𝐴 ((𝐹𝑞) ∩ 𝐴) = {𝑞}) → (∀𝑝𝐴𝑞𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} ∧ ∀𝑝𝐴𝑞𝐴 ((𝐹𝑞) ∩ 𝐴) = {𝑞}))
166, 15mpdan 686 . . . . 5 (∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} → (∀𝑝𝐴𝑞𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} ∧ ∀𝑝𝐴𝑞𝐴 ((𝐹𝑞) ∩ 𝐴) = {𝑞}))
17 r19.26-2 3140 . . . . 5 (∀𝑝𝐴𝑞𝐴 (((𝐹𝑝) ∩ 𝐴) = {𝑝} ∧ ((𝐹𝑞) ∩ 𝐴) = {𝑞}) ↔ (∀𝑝𝐴𝑞𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} ∧ ∀𝑝𝐴𝑞𝐴 ((𝐹𝑞) ∩ 𝐴) = {𝑞}))
1816, 17sylibr 234 . . . 4 (∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} → ∀𝑝𝐴𝑞𝐴 (((𝐹𝑝) ∩ 𝐴) = {𝑝} ∧ ((𝐹𝑞) ∩ 𝐴) = {𝑞}))
19 ineq1 4228 . . . . . . 7 ((𝐹𝑝) = (𝐹𝑞) → ((𝐹𝑝) ∩ 𝐴) = ((𝐹𝑞) ∩ 𝐴))
20 eqeq1 2738 . . . . . . . . 9 (((𝐹𝑝) ∩ 𝐴) = {𝑝} → (((𝐹𝑝) ∩ 𝐴) = ((𝐹𝑞) ∩ 𝐴) ↔ {𝑝} = ((𝐹𝑞) ∩ 𝐴)))
21 eqcom 2741 . . . . . . . . 9 ({𝑝} = ((𝐹𝑞) ∩ 𝐴) ↔ ((𝐹𝑞) ∩ 𝐴) = {𝑝})
2220, 21bitrdi 287 . . . . . . . 8 (((𝐹𝑝) ∩ 𝐴) = {𝑝} → (((𝐹𝑝) ∩ 𝐴) = ((𝐹𝑞) ∩ 𝐴) ↔ ((𝐹𝑞) ∩ 𝐴) = {𝑝}))
23 eqeq1 2738 . . . . . . . . 9 (((𝐹𝑞) ∩ 𝐴) = {𝑞} → (((𝐹𝑞) ∩ 𝐴) = {𝑝} ↔ {𝑞} = {𝑝}))
24 eqcom 2741 . . . . . . . . . 10 ({𝑞} = {𝑝} ↔ {𝑝} = {𝑞})
25 vex 3486 . . . . . . . . . . 11 𝑝 ∈ V
26 sneqbg 4868 . . . . . . . . . . 11 (𝑝 ∈ V → ({𝑝} = {𝑞} ↔ 𝑝 = 𝑞))
2725, 26ax-mp 5 . . . . . . . . . 10 ({𝑝} = {𝑞} ↔ 𝑝 = 𝑞)
2824, 27bitri 275 . . . . . . . . 9 ({𝑞} = {𝑝} ↔ 𝑝 = 𝑞)
2923, 28bitrdi 287 . . . . . . . 8 (((𝐹𝑞) ∩ 𝐴) = {𝑞} → (((𝐹𝑞) ∩ 𝐴) = {𝑝} ↔ 𝑝 = 𝑞))
3022, 29sylan9bb 509 . . . . . . 7 ((((𝐹𝑝) ∩ 𝐴) = {𝑝} ∧ ((𝐹𝑞) ∩ 𝐴) = {𝑞}) → (((𝐹𝑝) ∩ 𝐴) = ((𝐹𝑞) ∩ 𝐴) ↔ 𝑝 = 𝑞))
3119, 30imbitrid 244 . . . . . 6 ((((𝐹𝑝) ∩ 𝐴) = {𝑝} ∧ ((𝐹𝑞) ∩ 𝐴) = {𝑞}) → ((𝐹𝑝) = (𝐹𝑞) → 𝑝 = 𝑞))
3231ralimi 3085 . . . . 5 (∀𝑞𝐴 (((𝐹𝑝) ∩ 𝐴) = {𝑝} ∧ ((𝐹𝑞) ∩ 𝐴) = {𝑞}) → ∀𝑞𝐴 ((𝐹𝑝) = (𝐹𝑞) → 𝑝 = 𝑞))
3332ralimi 3085 . . . 4 (∀𝑝𝐴𝑞𝐴 (((𝐹𝑝) ∩ 𝐴) = {𝑝} ∧ ((𝐹𝑞) ∩ 𝐴) = {𝑞}) → ∀𝑝𝐴𝑞𝐴 ((𝐹𝑝) = (𝐹𝑞) → 𝑝 = 𝑞))
3418, 33syl 17 . . 3 (∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} → ∀𝑝𝐴𝑞𝐴 ((𝐹𝑝) = (𝐹𝑞) → 𝑝 = 𝑞))
3534anim2i 616 . 2 ((𝐹:𝐴𝐽 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → (𝐹:𝐴𝐽 ∧ ∀𝑝𝐴𝑞𝐴 ((𝐹𝑝) = (𝐹𝑞) → 𝑝 = 𝑞)))
36 dff13 7290 . 2 (𝐹:𝐴1-1𝐽 ↔ (𝐹:𝐴𝐽 ∧ ∀𝑝𝐴𝑞𝐴 ((𝐹𝑝) = (𝐹𝑞) → 𝑝 = 𝑞)))
3735, 36sylibr 234 1 ((𝐹:𝐴𝐽 ∧ ∀𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝}) → 𝐹:𝐴1-1𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wcel 2103  wral 3063  Vcvv 3482  cin 3969  {csn 4648  wf 6568  1-1wf1 6569  cfv 6572
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 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fv 6580
This theorem is referenced by:  fvineqsneu  37326  pibt2  37332
  Copyright terms: Public domain W3C validator