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Mirrors > Home > MPE Home > Th. List > Mathboxes > f1resrcmplf1d | Structured version Visualization version GIF version |
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.) |
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))) = ∅) |
Ref | Expression |
---|---|
f1resrcmplf1d | ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵) |
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 32377 | . . . . . 6 ⊢ (((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) | |
4 | 2, 3 | sylan 582 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) |
5 | 4 | ex 415 | . . . 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 32378 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ (𝐴 ∖ 𝐶)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))) |
10 | incom 4171 | . . . . . 6 ⊢ ((𝐹 “ 𝐶) ∩ (𝐹 “ (𝐴 ∖ 𝐶))) = ((𝐹 “ (𝐴 ∖ 𝐶)) ∩ (𝐹 “ 𝐶)) | |
11 | 10, 8 | syl5eqr 2869 | . . . . 5 ⊢ (𝜑 → ((𝐹 “ (𝐴 ∖ 𝐶)) ∩ (𝐹 “ 𝐶)) = ∅) |
12 | 7, 6, 1, 11 | f1resrcmplf1dlem 32378 | . . . 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 32377 | . . . . . 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 582 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐴 ∖ 𝐶) ∧ 𝑦 ∈ (𝐴 ∖ 𝐶))) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) |
16 | 15 | ex 415 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∖ 𝐶) ∧ 𝑦 ∈ (𝐴 ∖ 𝐶)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))) |
17 | 5, 9, 12, 16 | prsrcmpltd 32366 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))) |
18 | 17 | ralrimivv 3189 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) |
19 | dff13 7006 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))) | |
20 | 1, 18, 19 | sylanbrc 585 | 1 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3137 ∖ cdif 3926 ∩ cin 3928 ⊆ wss 3929 ∅c0 4284 ↾ cres 5550 “ cima 5551 ⟶wf 6344 –1-1→wf1 6345 ‘cfv 6348 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fv 6356 |
This theorem is referenced by: f1resfz0f1d 32380 |
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