Theorem fvineqsnf1 37433

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 37432 for a proof of this fact. (Contributed by ML, 23-Mar-2021.)

Assertion

Ref	Expression
fvineqsnf1	⊢ ((𝐹:𝐴⟶𝐽 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → 𝐹:𝐴–1-1→𝐽)

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.

Step	Hyp	Ref	Expression
1		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑝 = 𝑞 → (𝐹‘𝑝) = (𝐹‘𝑞))
2	1	ineq1d 4199	. . . . . . . . 9 ⊢ (𝑝 = 𝑞 → ((𝐹‘𝑝) ∩ 𝐴) = ((𝐹‘𝑞) ∩ 𝐴))
3		sneq 4616	. . . . . . . . 9 ⊢ (𝑝 = 𝑞 → {𝑝} = {𝑞})
4	2, 3	eqeq12d 2752	. . . . . . . 8 ⊢ (𝑝 = 𝑞 → (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ↔ ((𝐹‘𝑞) ∩ 𝐴) = {𝑞}))
5	4	cbvralvw 3224	. . . . . . 7 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ↔ ∀𝑞 ∈ 𝐴 ((𝐹‘𝑞) ∩ 𝐴) = {𝑞})
6	5	biimpi 216	. . . . . 6 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → ∀𝑞 ∈ 𝐴 ((𝐹‘𝑞) ∩ 𝐴) = {𝑞})
7		ax-5 1910	. . . . . . . 8 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → ∀𝑞∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝})
8		alral 3066	. . . . . . . 8 ⊢ (∀𝑞∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → ∀𝑞 ∈ 𝐴 ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝})
9		ralcom 3274	. . . . . . . . 9 ⊢ (∀𝑞 ∈ 𝐴 ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ↔ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝})
10	9	biimpi 216	. . . . . . . 8 ⊢ (∀𝑞 ∈ 𝐴 ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝})
11	7, 8, 10	3syl 18	. . . . . . 7 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝})
12		ax-5 1910	. . . . . . . 8 ⊢ (∀𝑞 ∈ 𝐴 ((𝐹‘𝑞) ∩ 𝐴) = {𝑞} → ∀𝑝∀𝑞 ∈ 𝐴 ((𝐹‘𝑞) ∩ 𝐴) = {𝑞})
13		alral 3066	. . . . . . . 8 ⊢ (∀𝑝∀𝑞 ∈ 𝐴 ((𝐹‘𝑞) ∩ 𝐴) = {𝑞} → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((𝐹‘𝑞) ∩ 𝐴) = {𝑞})
14	12, 13	syl 17	. . . . . . 7 ⊢ (∀𝑞 ∈ 𝐴 ((𝐹‘𝑞) ∩ 𝐴) = {𝑞} → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((𝐹‘𝑞) ∩ 𝐴) = {𝑞})
15	11, 14	anim12i 613	. . . . . 6 ⊢ ((∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ∧ ∀𝑞 ∈ 𝐴 ((𝐹‘𝑞) ∩ 𝐴) = {𝑞}) → (∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((𝐹‘𝑞) ∩ 𝐴) = {𝑞}))
16	6, 15	mpdan 687	. . . . 5 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((𝐹‘𝑞) ∩ 𝐴) = {𝑞}))
17		r19.26-2 3126	. . . . 5 ⊢ (∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ∧ ((𝐹‘𝑞) ∩ 𝐴) = {𝑞}) ↔ (∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((𝐹‘𝑞) ∩ 𝐴) = {𝑞}))
18	16, 17	sylibr 234	. . . 4 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ∧ ((𝐹‘𝑞) ∩ 𝐴) = {𝑞}))
19		ineq1 4193	. . . . . . 7 ⊢ ((𝐹‘𝑝) = (𝐹‘𝑞) → ((𝐹‘𝑝) ∩ 𝐴) = ((𝐹‘𝑞) ∩ 𝐴))
20		eqeq1 2740	. . . . . . . . 9 ⊢ (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (((𝐹‘𝑝) ∩ 𝐴) = ((𝐹‘𝑞) ∩ 𝐴) ↔ {𝑝} = ((𝐹‘𝑞) ∩ 𝐴)))
21		eqcom 2743	. . . . . . . . 9 ⊢ ({𝑝} = ((𝐹‘𝑞) ∩ 𝐴) ↔ ((𝐹‘𝑞) ∩ 𝐴) = {𝑝})
22	20, 21	bitrdi 287	. . . . . . . 8 ⊢ (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (((𝐹‘𝑝) ∩ 𝐴) = ((𝐹‘𝑞) ∩ 𝐴) ↔ ((𝐹‘𝑞) ∩ 𝐴) = {𝑝}))
23		eqeq1 2740	. . . . . . . . 9 ⊢ (((𝐹‘𝑞) ∩ 𝐴) = {𝑞} → (((𝐹‘𝑞) ∩ 𝐴) = {𝑝} ↔ {𝑞} = {𝑝}))
24		eqcom 2743	. . . . . . . . . 10 ⊢ ({𝑞} = {𝑝} ↔ {𝑝} = {𝑞})
25		vex 3468	. . . . . . . . . . 11 ⊢ 𝑝 ∈ V
26		sneqbg 4824	. . . . . . . . . . 11 ⊢ (𝑝 ∈ V → ({𝑝} = {𝑞} ↔ 𝑝 = 𝑞))
27	25, 26	ax-mp 5	. . . . . . . . . 10 ⊢ ({𝑝} = {𝑞} ↔ 𝑝 = 𝑞)
28	24, 27	bitri 275	. . . . . . . . 9 ⊢ ({𝑞} = {𝑝} ↔ 𝑝 = 𝑞)
29	23, 28	bitrdi 287	. . . . . . . 8 ⊢ (((𝐹‘𝑞) ∩ 𝐴) = {𝑞} → (((𝐹‘𝑞) ∩ 𝐴) = {𝑝} ↔ 𝑝 = 𝑞))
30	22, 29	sylan9bb 509	. . . . . . 7 ⊢ ((((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ∧ ((𝐹‘𝑞) ∩ 𝐴) = {𝑞}) → (((𝐹‘𝑝) ∩ 𝐴) = ((𝐹‘𝑞) ∩ 𝐴) ↔ 𝑝 = 𝑞))
31	19, 30	imbitrid 244	. . . . . 6 ⊢ ((((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ∧ ((𝐹‘𝑞) ∩ 𝐴) = {𝑞}) → ((𝐹‘𝑝) = (𝐹‘𝑞) → 𝑝 = 𝑞))
32	31	ralimi 3074	. . . . 5 ⊢ (∀𝑞 ∈ 𝐴 (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ∧ ((𝐹‘𝑞) ∩ 𝐴) = {𝑞}) → ∀𝑞 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑞) → 𝑝 = 𝑞))
33	32	ralimi 3074	. . . 4 ⊢ (∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ∧ ((𝐹‘𝑞) ∩ 𝐴) = {𝑞}) → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑞) → 𝑝 = 𝑞))
34	18, 33	syl 17	. . 3 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑞) → 𝑝 = 𝑞))
35	34	anim2i 617	. 2 ⊢ ((𝐹:𝐴⟶𝐽 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝐹:𝐴⟶𝐽 ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑞) → 𝑝 = 𝑞)))
36		dff13 7252	. 2 ⊢ (𝐹:𝐴–1-1→𝐽 ↔ (𝐹:𝐴⟶𝐽 ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑞) → 𝑝 = 𝑞)))
37	35, 36	sylibr 234	1 ⊢ ((𝐹:𝐴⟶𝐽 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → 𝐹:𝐴–1-1→𝐽)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∀wral 3052  Vcvv 3464   ∩ cin 3930  {csn 4606  ⟶wf 6532  –1-1→wf1 6533  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fv 6544

This theorem is referenced by:  fvineqsneu  37434  pibt2  37440