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Theorem dff15 34707
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 7265 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
2 iman 401 . . . . . 6 (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ¬ ((𝐹𝑥) = (𝐹𝑦) ∧ ¬ 𝑥 = 𝑦))
3 df-ne 2938 . . . . . . 7 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
43anbi2i 622 . . . . . 6 (((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦) ↔ ((𝐹𝑥) = (𝐹𝑦) ∧ ¬ 𝑥 = 𝑦))
52, 4xchbinxr 335 . . . . 5 (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ¬ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))
652ralbii 3125 . . . 4 (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 ¬ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))
7 ralnex2 3130 . . . 4 (∀𝑥𝐴𝑦𝐴 ¬ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦) ↔ ¬ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))
86, 7bitri 275 . . 3 (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ¬ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))
98anbi2i 622 . 2 ((𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ↔ (𝐹:𝐴𝐵 ∧ ¬ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)))
101, 9bitri 275 1 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ¬ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1534  wne 2937  wral 3058  wrex 3067  wf 6544  1-1wf1 6545  cfv 6548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fv 6556
This theorem is referenced by:  umgracycusgr  34764
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