MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dff14a Structured version   Visualization version   GIF version

Theorem dff14a 6690
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 6675 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
2 con34b 305 . . . . 5 (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (¬ 𝑥 = 𝑦 → ¬ (𝐹𝑥) = (𝐹𝑦)))
3 df-ne 2933 . . . . . . 7 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
43bicomi 214 . . . . . 6 𝑥 = 𝑦𝑥𝑦)
5 df-ne 2933 . . . . . . 7 ((𝐹𝑥) ≠ (𝐹𝑦) ↔ ¬ (𝐹𝑥) = (𝐹𝑦))
65bicomi 214 . . . . . 6 (¬ (𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑥) ≠ (𝐹𝑦))
74, 6imbi12i 339 . . . . 5 ((¬ 𝑥 = 𝑦 → ¬ (𝐹𝑥) = (𝐹𝑦)) ↔ (𝑥𝑦 → (𝐹𝑥) ≠ (𝐹𝑦)))
82, 7bitri 264 . . . 4 (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (𝑥𝑦 → (𝐹𝑥) ≠ (𝐹𝑦)))
982ralbii 3119 . . 3 (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≠ (𝐹𝑦)))
109anbi2i 732 . 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 383   = wceq 1632  wne 2932  wral 3050  wf 6045  1-1wf1 6046  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fv 6057
This theorem is referenced by:  dff14b  6691  pthdlem1  26872  nnfoctbdjlem  41175
  Copyright terms: Public domain W3C validator