Theorem f1resrcmplf1d 7310

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⟶1-1𝐵)
f1resrcmplf1d.4	⊢ (𝜑 → (𝐹 ↾ (𝐴 ∖ 𝐶)):(𝐴 ∖ 𝐶)⟶1-1𝐵)
f1resrcmplf1d.5	⊢ (𝜑 → ((𝐹 “ 𝐶) ∩ (𝐹 “ (𝐴 ∖ 𝐶))) = ∅)

Assertion

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

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⟶1-1𝐵)
3		f1resveqaeq 7308	. . . . . 6 ⊢ (((𝐹 ↾ 𝐶):𝐶⟶1-1𝐵 ∧ (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 4076	. . . . . . 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 7309	. . . . 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 4109	. . . . . . . 8 ⊢ ((𝐹 “ 𝐶) ∩ (𝐹 “ (𝐴 ∖ 𝐶))) = ((𝐹 “ (𝐴 ∖ 𝐶)) ∩ (𝐹 “ 𝐶))
22	21, 11	eqtr3id 2812	. . . . . . 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 7309	. . . . 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)⟶1-1𝐵)
30		f1resveqaeq 7308	. . . . . 6 ⊢ (((𝐹 ↾ (𝐴 ∖ 𝐶)):(𝐴 ∖ 𝐶)⟶1-1𝐵 ∧ (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 4436	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
34	33	ralrimivv 3207	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
35		dff13 7289	. 2 ⊢ (𝐹:𝐴⟶1-1𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
36	1, 34, 35	sylanbrc 595	1 ⊢ (𝜑 → 𝐹:𝐴⟶1-1𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092  ∅c0 4281   ↾ cres 5660   “ cima 5661  ⟶wf 6562  ⟶1-1wf1 6563  ‘cfv 6566

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fv 6574

This theorem is used by:  f1resfz0f1d  13967