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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 2939 . . . . . . 7 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
43bicomi 224 . . . . . 6 𝑥 = 𝑦𝑥𝑦)
5 df-ne 2939 . . . . . . 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 3126 . . 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 1537  wne 2938  wral 3059  wf 6559  1-1wf1 6560  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fv 6571
This theorem is referenced by:  dff14b  7291  f1ounsn  7292  pthdlem1  29799  fldhmf1  42072  nnfoctbdjlem  46411  isubgr3stgrlem4  47872
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