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Theorem dff14a 7290
Description: A one-to-one function in terms of different function values for different arguments. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
Assertion
Ref Expression
dff14a (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≠ (𝐹𝑦))))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem dff14a
StepHypRef Expression
1 dff13 7275 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
2 con34b 316 . . . . 5 (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (¬ 𝑥 = 𝑦 → ¬ (𝐹𝑥) = (𝐹𝑦)))
3 df-ne 2941 . . . . . . 7 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
43bicomi 224 . . . . . 6 𝑥 = 𝑦𝑥𝑦)
5 df-ne 2941 . . . . . . 7 ((𝐹𝑥) ≠ (𝐹𝑦) ↔ ¬ (𝐹𝑥) = (𝐹𝑦))
65bicomi 224 . . . . . 6 (¬ (𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑥) ≠ (𝐹𝑦))
74, 6imbi12i 350 . . . . 5 ((¬ 𝑥 = 𝑦 → ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ (𝑥𝑦 → (𝐹𝑥) ≠ (𝐹𝑦)))
82, 7bitri 275 . . . 4 (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (𝑥𝑦 → (𝐹𝑥) ≠ (𝐹𝑦)))
982ralbii 3128 . . 3 (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≠ (𝐹𝑦)))
109anbi2i 623 . 2 ((𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≠ (𝐹𝑦))))
111, 10bitri 275 1 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≠ (𝐹𝑦))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wne 2940  wral 3061  wf 6557  1-1wf1 6558  cfv 6561
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fv 6569
This theorem is referenced by:  dff14b  7291  f1ounsn  7292  resf1extb  7956  pthdlem1  29786  fldhmf1  42091  nnfoctbdjlem  46470  isubgr3stgrlem4  47936
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