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Theorem dff14a 6481
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 6466 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
2 con34b 306 . . . . 5 (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (¬ 𝑥 = 𝑦 → ¬ (𝐹𝑥) = (𝐹𝑦)))
3 df-ne 2791 . . . . . . 7 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
43bicomi 214 . . . . . 6 𝑥 = 𝑦𝑥𝑦)
5 df-ne 2791 . . . . . . 7 ((𝐹𝑥) ≠ (𝐹𝑦) ↔ ¬ (𝐹𝑥) = (𝐹𝑦))
65bicomi 214 . . . . . 6 (¬ (𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑥) ≠ (𝐹𝑦))
74, 6imbi12i 340 . . . . 5 ((¬ 𝑥 = 𝑦 → ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ (𝑥𝑦 → (𝐹𝑥) ≠ (𝐹𝑦)))
82, 7bitri 264 . . . 4 (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (𝑥𝑦 → (𝐹𝑥) ≠ (𝐹𝑦)))
982ralbii 2975 . . 3 (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≠ (𝐹𝑦)))
109anbi2i 729 . 2 ((𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≠ (𝐹𝑦))))
111, 10bitri 264 1 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≠ (𝐹𝑦))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wne 2790  wral 2907  wf 5843  1-1wf1 5844  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fv 5855
This theorem is referenced by:  dff14b  6482  pthdlem1  26531  nnfoctbdjlem  39979
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