Theorem fvineqsnf1 38426

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 38425 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 6885	. . . . . . . . . 10 ⊢ (𝑝 = 𝑞 → (𝐹‘𝑝) = (𝐹‘𝑞))
2	1	ineq1d 4116	. . . . . . . . 9 ⊢ (𝑝 = 𝑞 → ((𝐹‘𝑝) ∩ 𝐴) = ((𝐹‘𝑞) ∩ 𝐴))
3		sneq 4590	. . . . . . . . 9 ⊢ (𝑝 = 𝑞 → {𝑝} = {𝑞})
4	2, 3	eqeq12d 2776	. . . . . . . 8 ⊢ (𝑝 = 𝑞 → (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ↔ ((𝐹‘𝑞) ∩ 𝐴) = {𝑞}))
5	4	cbvralvw 3240	. . . . . . 7 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ↔ ∀𝑞 ∈ 𝐴 ((𝐹‘𝑞) ∩ 𝐴) = {𝑞})
6	5	biimpi 219	. . . . . 6 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → ∀𝑞 ∈ 𝐴 ((𝐹‘𝑞) ∩ 𝐴) = {𝑞})
7		ax-5 1942	. . . . . . . 8 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → ∀𝑞∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝})
8		alral 3091	. . . . . . . 8 ⊢ (∀𝑞∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → ∀𝑞 ∈ 𝐴 ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝})
9		ralcom 3290	. . . . . . . . 9 ⊢ (∀𝑞 ∈ 𝐴 ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ↔ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝})
10	9	biimpi 219	. . . . . . . 8 ⊢ (∀𝑞 ∈ 𝐴 ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝})
11	7, 8, 10	3syl 19	. . . . . . 7 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝})
12		ax-5 1942	. . . . . . . 8 ⊢ (∀𝑞 ∈ 𝐴 ((𝐹‘𝑞) ∩ 𝐴) = {𝑞} → ∀𝑝∀𝑞 ∈ 𝐴 ((𝐹‘𝑞) ∩ 𝐴) = {𝑞})
13		alral 3091	. . . . . . . 8 ⊢ (∀𝑝∀𝑞 ∈ 𝐴 ((𝐹‘𝑞) ∩ 𝐴) = {𝑞} → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((𝐹‘𝑞) ∩ 𝐴) = {𝑞})
14	12, 13	syl 18	. . . . . . 7 ⊢ (∀𝑞 ∈ 𝐴 ((𝐹‘𝑞) ∩ 𝐴) = {𝑞} → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((𝐹‘𝑞) ∩ 𝐴) = {𝑞})
15	11, 14	anim12i 625	. . . . . 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 700	. . . . 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 3147	. . . . 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 237	. . . 4 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ∧ ((𝐹‘𝑞) ∩ 𝐴) = {𝑞}))
19		ineq1 4110	. . . . . . 7 ⊢ ((𝐹‘𝑝) = (𝐹‘𝑞) → ((𝐹‘𝑝) ∩ 𝐴) = ((𝐹‘𝑞) ∩ 𝐴))
20		eqeq1 2764	. . . . . . . . 9 ⊢ (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (((𝐹‘𝑝) ∩ 𝐴) = ((𝐹‘𝑞) ∩ 𝐴) ↔ {𝑝} = ((𝐹‘𝑞) ∩ 𝐴)))
21		eqcom 2767	. . . . . . . . 9 ⊢ ({𝑝} = ((𝐹‘𝑞) ∩ 𝐴) ↔ ((𝐹‘𝑞) ∩ 𝐴) = {𝑝})
22	20, 21	bitrdi 290	. . . . . . . 8 ⊢ (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (((𝐹‘𝑝) ∩ 𝐴) = ((𝐹‘𝑞) ∩ 𝐴) ↔ ((𝐹‘𝑞) ∩ 𝐴) = {𝑝}))
23		eqeq1 2764	. . . . . . . . 9 ⊢ (((𝐹‘𝑞) ∩ 𝐴) = {𝑞} → (((𝐹‘𝑞) ∩ 𝐴) = {𝑝} ↔ {𝑞} = {𝑝}))
24		eqcom 2767	. . . . . . . . . 10 ⊢ ({𝑞} = {𝑝} ↔ {𝑝} = {𝑞})
25		vex 3454	. . . . . . . . . . 11 ⊢ 𝑝 ∈ V
26		sneqbg 4797	. . . . . . . . . . 11 ⊢ (𝑝 ∈ V → ({𝑝} = {𝑞} ↔ 𝑝 = 𝑞))
27	25, 26	ax-mp 5	. . . . . . . . . 10 ⊢ ({𝑝} = {𝑞} ↔ 𝑝 = 𝑞)
28	24, 27	bitri 278	. . . . . . . . 9 ⊢ ({𝑞} = {𝑝} ↔ 𝑝 = 𝑞)
29	23, 28	bitrdi 290	. . . . . . . 8 ⊢ (((𝐹‘𝑞) ∩ 𝐴) = {𝑞} → (((𝐹‘𝑞) ∩ 𝐴) = {𝑝} ↔ 𝑝 = 𝑞))
30	22, 29	sylan9bb 519	. . . . . . 7 ⊢ ((((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ∧ ((𝐹‘𝑞) ∩ 𝐴) = {𝑞}) → (((𝐹‘𝑝) ∩ 𝐴) = ((𝐹‘𝑞) ∩ 𝐴) ↔ 𝑝 = 𝑞))
31	19, 30	imbitrid 247	. . . . . 6 ⊢ ((((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ∧ ((𝐹‘𝑞) ∩ 𝐴) = {𝑞}) → ((𝐹‘𝑝) = (𝐹‘𝑞) → 𝑝 = 𝑞))
32	31	ralimi 3099	. . . . 5 ⊢ (∀𝑞 ∈ 𝐴 (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ∧ ((𝐹‘𝑞) ∩ 𝐴) = {𝑞}) → ∀𝑞 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑞) → 𝑝 = 𝑞))
33	32	ralimi 3099	. . . 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 18	. . 3 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑞) → 𝑝 = 𝑞))
35	34	anim2i 629	. 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 7258	. 2 ⊢ (𝐹:𝐴–1-1→𝐽 ↔ (𝐹:𝐴⟶𝐽 ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑞) → 𝑝 = 𝑞)))
37	35, 36	sylibr 237	1 ⊢ ((𝐹:𝐴⟶𝐽 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → 𝐹:𝐴–1-1→𝐽)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567   = wceq 1569   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ∩ cin 4089  {csn 4584  ⟶wf 6534  –1-1→wf1 6535  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fv 6546

This theorem is used by:  fvineqsneu  38427  pibt2  38433