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Theorem fvineqsnf1 37376
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 37375 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 6920 . . . . . . . . . 10 (𝑝 = 𝑞 → (𝐹𝑝) = (𝐹𝑞))
21ineq1d 4240 . . . . . . . . 9 (𝑝 = 𝑞 → ((𝐹𝑝) ∩ 𝐴) = ((𝐹𝑞) ∩ 𝐴))
3 sneq 4658 . . . . . . . . 9 (𝑝 = 𝑞 → {𝑝} = {𝑞})
42, 3eqeq12d 2756 . . . . . . . 8 (𝑝 = 𝑞 → (((𝐹𝑝) ∩ 𝐴) = {𝑝} ↔ ((𝐹𝑞) ∩ 𝐴) = {𝑞}))
54cbvralvw 3243 . . . . . . 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 3081 . . . . . . . 8 (∀𝑞𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝} → ∀𝑞𝐴𝑝𝐴 ((𝐹𝑝) ∩ 𝐴) = {𝑝})
9 ralcom 3295 . . . . . . . . 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 3081 . . . . . . . 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 3144 . . . . 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 4234 . . . . . . 7 ((𝐹𝑝) = (𝐹𝑞) → ((𝐹𝑝) ∩ 𝐴) = ((𝐹𝑞) ∩ 𝐴))
20 eqeq1 2744 . . . . . . . . 9 (((𝐹𝑝) ∩ 𝐴) = {𝑝} → (((𝐹𝑝) ∩ 𝐴) = ((𝐹𝑞) ∩ 𝐴) ↔ {𝑝} = ((𝐹𝑞) ∩ 𝐴)))
21 eqcom 2747 . . . . . . . . 9 ({𝑝} = ((𝐹𝑞) ∩ 𝐴) ↔ ((𝐹𝑞) ∩ 𝐴) = {𝑝})
2220, 21bitrdi 287 . . . . . . . 8 (((𝐹𝑝) ∩ 𝐴) = {𝑝} → (((𝐹𝑝) ∩ 𝐴) = ((𝐹𝑞) ∩ 𝐴) ↔ ((𝐹𝑞) ∩ 𝐴) = {𝑝}))
23 eqeq1 2744 . . . . . . . . 9 (((𝐹𝑞) ∩ 𝐴) = {𝑞} → (((𝐹𝑞) ∩ 𝐴) = {𝑝} ↔ {𝑞} = {𝑝}))
24 eqcom 2747 . . . . . . . . . 10 ({𝑞} = {𝑝} ↔ {𝑝} = {𝑞})
25 vex 3492 . . . . . . . . . . 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 3089 . . . . 5 (∀𝑞𝐴 (((𝐹𝑝) ∩ 𝐴) = {𝑝} ∧ ((𝐹𝑞) ∩ 𝐴) = {𝑞}) → ∀𝑞𝐴 ((𝐹𝑝) = (𝐹𝑞) → 𝑝 = 𝑞))
3332ralimi 3089 . . . 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 7292 . 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 2108  wral 3067  Vcvv 3488  cin 3975  {csn 4648  wf 6569  1-1wf1 6570  cfv 6573
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-pr 5447
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 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fv 6581
This theorem is referenced by:  fvineqsneu  37377  pibt2  37383
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