Mathbox for BTernaryTau < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dff15 Structured version   Visualization version   GIF version

Theorem dff15 32361
 Description: A one-to-one function in terms of different arguments never having the same function value. (Contributed by BTernaryTau, 24-Oct-2023.)
Assertion
Ref Expression
dff15 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ¬ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem dff15
StepHypRef Expression
1 dff13 6987 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
2 iman 405 . . . . . 6 (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ¬ ((𝐹𝑥) = (𝐹𝑦) ∧ ¬ 𝑥 = 𝑦))
3 df-ne 3008 . . . . . . 7 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
43anbi2i 625 . . . . . 6 (((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦) ↔ ((𝐹𝑥) = (𝐹𝑦) ∧ ¬ 𝑥 = 𝑦))
52, 4xchbinxr 338 . . . . 5 (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ¬ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))
652ralbii 3154 . . . 4 (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 ¬ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))
7 ralnex2 3248 . . . 4 (∀𝑥𝐴𝑦𝐴 ¬ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦) ↔ ¬ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))
86, 7bitri 278 . . 3 (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ¬ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))
98anbi2i 625 . 2 ((𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ↔ (𝐹:𝐴𝐵 ∧ ¬ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)))
101, 9bitri 278 1 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ¬ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ≠ wne 3007  ∀wral 3126  ∃wrex 3127  ⟶wf 6324  –1-1→wf1 6325  ‘cfv 6328 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fv 6336 This theorem is referenced by:  umgracycusgr  32409
 Copyright terms: Public domain W3C validator