Theorem f1resrcmplf1d 7275

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 7273	. . . . . 6 ⊢ (((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
4	2, 3	sylan 592	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
5	4	ex 418	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
6		f1resrcmplf1d.1	. . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
7	6	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ (𝐴 ∖ 𝐶)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝐶 ⊆ 𝐴)
8		difssd 4091	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ 𝐴)
9	8	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ (𝐴 ∖ 𝐶)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐴 ∖ 𝐶) ⊆ 𝐴)
10	1	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ (𝐴 ∖ 𝐶)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝐹:𝐴⟶𝐵)
11		f1resrcmplf1d.5	. . . . . . 7 ⊢ (𝜑 → ((𝐹 “ 𝐶) ∩ (𝐹 “ (𝐴 ∖ 𝐶))) = ∅)
12	11	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ (𝐴 ∖ 𝐶)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → ((𝐹 “ 𝐶) ∩ (𝐹 “ (𝐴 ∖ 𝐶))) = ∅)
13		simp2l 1218	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ (𝐴 ∖ 𝐶)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑥 ∈ 𝐶)
14		simp2r 1219	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ (𝐴 ∖ 𝐶)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑦 ∈ (𝐴 ∖ 𝐶))
15		simp3 1156	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ (𝐴 ∖ 𝐶)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐹‘𝑥) = (𝐹‘𝑦))
16	7, 9, 10, 12, 13, 14, 15	f1resrcmplf1dlem 7274	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ (𝐴 ∖ 𝐶)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑥 = 𝑦)
17	16	3exp 1137	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ (𝐴 ∖ 𝐶)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
18	8	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐴 ∖ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐴 ∖ 𝐶) ⊆ 𝐴)
19	6	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐴 ∖ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝐶 ⊆ 𝐴)
20	1	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐴 ∖ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝐹:𝐴⟶𝐵)
21		incom 4162	. . . . . . . 8 ⊢ ((𝐹 “ 𝐶) ∩ (𝐹 “ (𝐴 ∖ 𝐶))) = ((𝐹 “ (𝐴 ∖ 𝐶)) ∩ (𝐹 “ 𝐶))
22	21, 11	eqtr3id 2814	. . . . . . 7 ⊢ (𝜑 → ((𝐹 “ (𝐴 ∖ 𝐶)) ∩ (𝐹 “ 𝐶)) = ∅)
23	22	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐴 ∖ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → ((𝐹 “ (𝐴 ∖ 𝐶)) ∩ (𝐹 “ 𝐶)) = ∅)
24		simp2l 1218	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐴 ∖ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑥 ∈ (𝐴 ∖ 𝐶))
25		simp2r 1219	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐴 ∖ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑦 ∈ 𝐶)
26		simp3 1156	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐴 ∖ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐹‘𝑥) = (𝐹‘𝑦))
27	18, 19, 20, 23, 24, 25, 26	f1resrcmplf1dlem 7274	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐴 ∖ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑥 = 𝑦)
28	27	3exp 1137	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∖ 𝐶) ∧ 𝑦 ∈ 𝐶) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
29		f1resrcmplf1d.4	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∖ 𝐶)):(𝐴 ∖ 𝐶)–1-1→𝐵)
30		f1resveqaeq 7273	. . . . . 6 ⊢ (((𝐹 ↾ (𝐴 ∖ 𝐶)):(𝐴 ∖ 𝐶)–1-1→𝐵 ∧ (𝑥 ∈ (𝐴 ∖ 𝐶) ∧ 𝑦 ∈ (𝐴 ∖ 𝐶))) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
31	29, 30	sylan 592	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐴 ∖ 𝐶) ∧ 𝑦 ∈ (𝐴 ∖ 𝐶))) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
32	31	ex 418	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∖ 𝐶) ∧ 𝑦 ∈ (𝐴 ∖ 𝐶)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
33	5, 17, 28, 32	prsrcmpltd 4440	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
34	33	ralrimivv 3208	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
35		dff13 7254	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
36	1, 34, 35	sylanbrc 595	1 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286   ↾ cres 5665   “ cima 5666  ⟶wf 6536  –1-1→wf1 6537  ‘cfv 6540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fv 6548

This theorem is used by:  f1resfz0f1d  13834