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Theorem dff15 35099
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 7276 . 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 2940 . . . . . . 7 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
43anbi2i 623 . . . . . 6 (((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦) ↔ ((𝐹𝑥) = (𝐹𝑦) ∧ ¬ 𝑥 = 𝑦))
52, 4xchbinxr 335 . . . . 5 (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ¬ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))
652ralbii 3127 . . . 4 (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 ¬ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))
7 ralnex2 3132 . . . 4 (∀𝑥𝐴𝑦𝐴 ¬ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦) ↔ ¬ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))
86, 7bitri 275 . . 3 (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ¬ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))
98anbi2i 623 . 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 206  wa 395   = wceq 1539  wne 2939  wral 3060  wrex 3069  wf 6556  1-1wf1 6557  cfv 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fv 6568
This theorem is referenced by:  umgracycusgr  35160
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