Theorem f1resrcmplf1d 35083

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 35081	. . . . . 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 412	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
6		f1resrcmplf1d.1	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
7		difssd 4102	. . . . 5 ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ 𝐴)
8		f1resrcmplf1d.5	. . . . 5 ⊢ (𝜑 → ((𝐹 “ 𝐶) ∩ (𝐹 “ (𝐴 ∖ 𝐶))) = ∅)
9	6, 7, 1, 8	f1resrcmplf1dlem 35082	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ (𝐴 ∖ 𝐶)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
10		incom 4174	. . . . . 6 ⊢ ((𝐹 “ 𝐶) ∩ (𝐹 “ (𝐴 ∖ 𝐶))) = ((𝐹 “ (𝐴 ∖ 𝐶)) ∩ (𝐹 “ 𝐶))
11	10, 8	eqtr3id 2779	. . . . 5 ⊢ (𝜑 → ((𝐹 “ (𝐴 ∖ 𝐶)) ∩ (𝐹 “ 𝐶)) = ∅)
12	7, 6, 1, 11	f1resrcmplf1dlem 35082	. . . 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 35081	. . . . . 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 412	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∖ 𝐶) ∧ 𝑦 ∈ (𝐴 ∖ 𝐶)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
17	5, 9, 12, 16	prsrcmpltd 35077	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
18	17	ralrimivv 3179	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
19		dff13 7231	. 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 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045   ∖ cdif 3913   ∩ cin 3915   ⊆ wss 3916  ∅c0 4298   ↾ cres 5642   “ cima 5643  ⟶wf 6509  –1-1→wf1 6510  ‘cfv 6513

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 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fv 6521

This theorem is referenced by:  f1resfz0f1d  35101