Theorem f1resrcmplf1d 35171

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 35169	. . . . . 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 4086	. . . . 5 ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ 𝐴)
8		f1resrcmplf1d.5	. . . . 5 ⊢ (𝜑 → ((𝐹 “ 𝐶) ∩ (𝐹 “ (𝐴 ∖ 𝐶))) = ∅)
9	6, 7, 1, 8	f1resrcmplf1dlem 35170	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ (𝐴 ∖ 𝐶)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
10		incom 4158	. . . . . 6 ⊢ ((𝐹 “ 𝐶) ∩ (𝐹 “ (𝐴 ∖ 𝐶))) = ((𝐹 “ (𝐴 ∖ 𝐶)) ∩ (𝐹 “ 𝐶))
11	10, 8	eqtr3id 2782	. . . . 5 ⊢ (𝜑 → ((𝐹 “ (𝐴 ∖ 𝐶)) ∩ (𝐹 “ 𝐶)) = ∅)
12	7, 6, 1, 11	f1resrcmplf1dlem 35170	. . . 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 35169	. . . . . 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 35165	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
18	17	ralrimivv 3174	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
19		dff13 7197	. 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 1541   ∈ wcel 2113  ∀wral 3048   ∖ cdif 3895   ∩ cin 3897   ⊆ wss 3898  ∅c0 4282   ↾ cres 5623   “ cima 5624  ⟶wf 6485  –1-1→wf1 6486  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fv 6497

This theorem is referenced by:  f1resfz0f1d  35230