Theorem f1resrcmplf1d 33059

Description: If a function's restriction to a subclass of its domain and its restriction to the relative complement of that subclass are both one-to-one, and if the ranges of those two restrictions are disjoint, then the function is itself one-to-one. (Contributed by BTernaryTau, 28-Sep-2023.)

Hypotheses

Ref	Expression
f1resrcmplf1d.1	⊢ (𝜑 → 𝐶 ⊆ 𝐴)
f1resrcmplf1d.2	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
f1resrcmplf1d.3	⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵)
f1resrcmplf1d.4	⊢ (𝜑 → (𝐹 ↾ (𝐴 ∖ 𝐶)):(𝐴 ∖ 𝐶)–1-1→𝐵)
f1resrcmplf1d.5	⊢ (𝜑 → ((𝐹 “ 𝐶) ∩ (𝐹 “ (𝐴 ∖ 𝐶))) = ∅)

Assertion

Ref	Expression
f1resrcmplf1d	⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)

Proof of Theorem f1resrcmplf1d

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1resrcmplf1d.2	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2		f1resrcmplf1d.3	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵)
3		f1resveqaeq 33057	. . . . . 6 ⊢ (((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
4	2, 3	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
5	4	ex 413	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
6		f1resrcmplf1d.1	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
7		difssd 4067	. . . . 5 ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ 𝐴)
8		f1resrcmplf1d.5	. . . . 5 ⊢ (𝜑 → ((𝐹 “ 𝐶) ∩ (𝐹 “ (𝐴 ∖ 𝐶))) = ∅)
9	6, 7, 1, 8	f1resrcmplf1dlem 33058	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ (𝐴 ∖ 𝐶)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
10		incom 4135	. . . . . 6 ⊢ ((𝐹 “ 𝐶) ∩ (𝐹 “ (𝐴 ∖ 𝐶))) = ((𝐹 “ (𝐴 ∖ 𝐶)) ∩ (𝐹 “ 𝐶))
11	10, 8	eqtr3id 2792	. . . . 5 ⊢ (𝜑 → ((𝐹 “ (𝐴 ∖ 𝐶)) ∩ (𝐹 “ 𝐶)) = ∅)
12	7, 6, 1, 11	f1resrcmplf1dlem 33058	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∖ 𝐶) ∧ 𝑦 ∈ 𝐶) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
13		f1resrcmplf1d.4	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∖ 𝐶)):(𝐴 ∖ 𝐶)–1-1→𝐵)
14		f1resveqaeq 33057	. . . . . 6 ⊢ (((𝐹 ↾ (𝐴 ∖ 𝐶)):(𝐴 ∖ 𝐶)–1-1→𝐵 ∧ (𝑥 ∈ (𝐴 ∖ 𝐶) ∧ 𝑦 ∈ (𝐴 ∖ 𝐶))) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
15	13, 14	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐴 ∖ 𝐶) ∧ 𝑦 ∈ (𝐴 ∖ 𝐶))) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
16	15	ex 413	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∖ 𝐶) ∧ 𝑦 ∈ (𝐴 ∖ 𝐶)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
17	5, 9, 12, 16	prsrcmpltd 33055	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
18	17	ralrimivv 3122	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
19		dff13 7128	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
20	1, 18, 19	sylanbrc 583	1 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ∖ cdif 3884   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256   ↾ cres 5591   “ cima 5592  ⟶wf 6429  –1-1→wf1 6430  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fv 6441

This theorem is referenced by:  f1resfz0f1d  33072