Theorem fvineqsnf1 35881

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 35880 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 6842	. . . . . . . . . 10 ⊢ (𝑝 = 𝑞 → (𝐹‘𝑝) = (𝐹‘𝑞))
2	1	ineq1d 4171	. . . . . . . . 9 ⊢ (𝑝 = 𝑞 → ((𝐹‘𝑝) ∩ 𝐴) = ((𝐹‘𝑞) ∩ 𝐴))
3		sneq 4596	. . . . . . . . 9 ⊢ (𝑝 = 𝑞 → {𝑝} = {𝑞})
4	2, 3	eqeq12d 2752	. . . . . . . 8 ⊢ (𝑝 = 𝑞 → (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ↔ ((𝐹‘𝑞) ∩ 𝐴) = {𝑞}))
5	4	cbvralvw 3225	. . . . . . 7 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ↔ ∀𝑞 ∈ 𝐴 ((𝐹‘𝑞) ∩ 𝐴) = {𝑞})
6	5	biimpi 215	. . . . . 6 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → ∀𝑞 ∈ 𝐴 ((𝐹‘𝑞) ∩ 𝐴) = {𝑞})
7		ax-5 1913	. . . . . . . 8 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → ∀𝑞∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝})
8		alral 3078	. . . . . . . 8 ⊢ (∀𝑞∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → ∀𝑞 ∈ 𝐴 ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝})
9		ralcom 3272	. . . . . . . . 9 ⊢ (∀𝑞 ∈ 𝐴 ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ↔ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝})
10	9	biimpi 215	. . . . . . . 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 1913	. . . . . . . 8 ⊢ (∀𝑞 ∈ 𝐴 ((𝐹‘𝑞) ∩ 𝐴) = {𝑞} → ∀𝑝∀𝑞 ∈ 𝐴 ((𝐹‘𝑞) ∩ 𝐴) = {𝑞})
13		alral 3078	. . . . . . . 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 685	. . . . 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 3135	. . . . 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 233	. . . 4 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ∧ ((𝐹‘𝑞) ∩ 𝐴) = {𝑞}))
19		ineq1 4165	. . . . . . 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 286	. . . . . . . 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 3449	. . . . . . . . . . 11 ⊢ 𝑝 ∈ V
26		sneqbg 4801	. . . . . . . . . . 11 ⊢ (𝑝 ∈ V → ({𝑝} = {𝑞} ↔ 𝑝 = 𝑞))
27	25, 26	ax-mp 5	. . . . . . . . . 10 ⊢ ({𝑝} = {𝑞} ↔ 𝑝 = 𝑞)
28	24, 27	bitri 274	. . . . . . . . 9 ⊢ ({𝑞} = {𝑝} ↔ 𝑝 = 𝑞)
29	23, 28	bitrdi 286	. . . . . . . 8 ⊢ (((𝐹‘𝑞) ∩ 𝐴) = {𝑞} → (((𝐹‘𝑞) ∩ 𝐴) = {𝑝} ↔ 𝑝 = 𝑞))
30	22, 29	sylan9bb 510	. . . . . . 7 ⊢ ((((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ∧ ((𝐹‘𝑞) ∩ 𝐴) = {𝑞}) → (((𝐹‘𝑝) ∩ 𝐴) = ((𝐹‘𝑞) ∩ 𝐴) ↔ 𝑝 = 𝑞))
31	19, 30	imbitrid 243	. . . . . 6 ⊢ ((((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ∧ ((𝐹‘𝑞) ∩ 𝐴) = {𝑞}) → ((𝐹‘𝑝) = (𝐹‘𝑞) → 𝑝 = 𝑞))
32	31	ralimi 3086	. . . . 5 ⊢ (∀𝑞 ∈ 𝐴 (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ∧ ((𝐹‘𝑞) ∩ 𝐴) = {𝑞}) → ∀𝑞 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑞) → 𝑝 = 𝑞))
33	32	ralimi 3086	. . . 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 7202	. 2 ⊢ (𝐹:𝐴–1-1→𝐽 ↔ (𝐹:𝐴⟶𝐽 ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑞) → 𝑝 = 𝑞)))
37	35, 36	sylibr 233	1 ⊢ ((𝐹:𝐴⟶𝐽 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → 𝐹:𝐴–1-1→𝐽)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1539   = wceq 1541   ∈ wcel 2106  ∀wral 3064  Vcvv 3445   ∩ cin 3909  {csn 4586  ⟶wf 6492  –1-1→wf1 6493  ‘cfv 6496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fv 6504

This theorem is referenced by:  fvineqsneu  35882  pibt2  35888