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Theorem f1resrcmplf1d 32581
 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 (𝜑𝐶𝐴)
f1resrcmplf1d.2 (𝜑𝐹:𝐴𝐵)
f1resrcmplf1d.3 (𝜑 → (𝐹𝐶):𝐶1-1𝐵)
f1resrcmplf1d.4 (𝜑 → (𝐹 ↾ (𝐴𝐶)):(𝐴𝐶)–1-1𝐵)
f1resrcmplf1d.5 (𝜑 → ((𝐹𝐶) ∩ (𝐹 “ (𝐴𝐶))) = ∅)
Assertion
Ref Expression
f1resrcmplf1d (𝜑𝐹:𝐴1-1𝐵)

Proof of Theorem f1resrcmplf1d
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1resrcmplf1d.2 . 2 (𝜑𝐹:𝐴𝐵)
2 f1resrcmplf1d.3 . . . . . 6 (𝜑 → (𝐹𝐶):𝐶1-1𝐵)
3 f1resveqaeq 32579 . . . . . 6 (((𝐹𝐶):𝐶1-1𝐵 ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
42, 3sylan 584 . . . . 5 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
54ex 417 . . . 4 (𝜑 → ((𝑥𝐶𝑦𝐶) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
6 f1resrcmplf1d.1 . . . . 5 (𝜑𝐶𝐴)
7 difssd 4039 . . . . 5 (𝜑 → (𝐴𝐶) ⊆ 𝐴)
8 f1resrcmplf1d.5 . . . . 5 (𝜑 → ((𝐹𝐶) ∩ (𝐹 “ (𝐴𝐶))) = ∅)
96, 7, 1, 8f1resrcmplf1dlem 32580 . . . 4 (𝜑 → ((𝑥𝐶𝑦 ∈ (𝐴𝐶)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
10 incom 4107 . . . . . 6 ((𝐹𝐶) ∩ (𝐹 “ (𝐴𝐶))) = ((𝐹 “ (𝐴𝐶)) ∩ (𝐹𝐶))
1110, 8syl5eqr 2808 . . . . 5 (𝜑 → ((𝐹 “ (𝐴𝐶)) ∩ (𝐹𝐶)) = ∅)
127, 6, 1, 11f1resrcmplf1dlem 32580 . . . 4 (𝜑 → ((𝑥 ∈ (𝐴𝐶) ∧ 𝑦𝐶) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
13 f1resrcmplf1d.4 . . . . . 6 (𝜑 → (𝐹 ↾ (𝐴𝐶)):(𝐴𝐶)–1-1𝐵)
14 f1resveqaeq 32579 . . . . . 6 (((𝐹 ↾ (𝐴𝐶)):(𝐴𝐶)–1-1𝐵 ∧ (𝑥 ∈ (𝐴𝐶) ∧ 𝑦 ∈ (𝐴𝐶))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
1513, 14sylan 584 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (𝐴𝐶) ∧ 𝑦 ∈ (𝐴𝐶))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
1615ex 417 . . . 4 (𝜑 → ((𝑥 ∈ (𝐴𝐶) ∧ 𝑦 ∈ (𝐴𝐶)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
175, 9, 12, 16prsrcmpltd 32568 . . 3 (𝜑 → ((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
1817ralrimivv 3120 . 2 (𝜑 → ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
19 dff13 7006 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
201, 18, 19sylanbrc 587 1 (𝜑𝐹:𝐴1-1𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∀wral 3071   ∖ cdif 3856   ∩ cin 3858   ⊆ wss 3859  ∅c0 4226   ↾ cres 5527   “ cima 5528  ⟶wf 6332  –1-1→wf1 6333  ‘cfv 6336 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fv 6344 This theorem is referenced by:  f1resfz0f1d  32582
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