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Theorem nf1const 7324
Description: A constant function from at least two elements is not one-to-one. (Contributed by AV, 30-Mar-2024.)
Assertion
Ref Expression
nf1const ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → ¬ 𝐹:𝐴1-1𝐶)

Proof of Theorem nf1const
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1135 . . . . 5 ((𝑋𝐴𝑌𝐴𝑋𝑌) → 𝑋𝐴)
2 simp2 1136 . . . . 5 ((𝑋𝐴𝑌𝐴𝑋𝑌) → 𝑌𝐴)
3 fvconst 7184 . . . . . . . 8 ((𝐹:𝐴⟶{𝐵} ∧ 𝑋𝐴) → (𝐹𝑋) = 𝐵)
41, 3sylan2 593 . . . . . . 7 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (𝐹𝑋) = 𝐵)
5 fvconst 7184 . . . . . . . 8 ((𝐹:𝐴⟶{𝐵} ∧ 𝑌𝐴) → (𝐹𝑌) = 𝐵)
62, 5sylan2 593 . . . . . . 7 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (𝐹𝑌) = 𝐵)
74, 6eqtr4d 2778 . . . . . 6 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (𝐹𝑋) = (𝐹𝑌))
8 neneq 2944 . . . . . . . 8 (𝑋𝑌 → ¬ 𝑋 = 𝑌)
983ad2ant3 1134 . . . . . . 7 ((𝑋𝐴𝑌𝐴𝑋𝑌) → ¬ 𝑋 = 𝑌)
109adantl 481 . . . . . 6 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → ¬ 𝑋 = 𝑌)
117, 10jcnd 163 . . . . 5 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → ¬ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))
12 fveqeq2 6916 . . . . . . . 8 (𝑥 = 𝑋 → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑋) = (𝐹𝑦)))
13 eqeq1 2739 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 = 𝑦𝑋 = 𝑦))
1412, 13imbi12d 344 . . . . . . 7 (𝑥 = 𝑋 → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦)))
1514notbid 318 . . . . . 6 (𝑥 = 𝑋 → (¬ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ¬ ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦)))
16 fveq2 6907 . . . . . . . . 9 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
1716eqeq2d 2746 . . . . . . . 8 (𝑦 = 𝑌 → ((𝐹𝑋) = (𝐹𝑦) ↔ (𝐹𝑋) = (𝐹𝑌)))
18 eqeq2 2747 . . . . . . . 8 (𝑦 = 𝑌 → (𝑋 = 𝑦𝑋 = 𝑌))
1917, 18imbi12d 344 . . . . . . 7 (𝑦 = 𝑌 → (((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦) ↔ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))
2019notbid 318 . . . . . 6 (𝑦 = 𝑌 → (¬ ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦) ↔ ¬ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))
2115, 20rspc2ev 3635 . . . . 5 ((𝑋𝐴𝑌𝐴 ∧ ¬ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)) → ∃𝑥𝐴𝑦𝐴 ¬ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
221, 2, 11, 21syl2an23an 1422 . . . 4 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → ∃𝑥𝐴𝑦𝐴 ¬ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
23 rexnal2 3133 . . . 4 (∃𝑥𝐴𝑦𝐴 ¬ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ¬ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
2422, 23sylib 218 . . 3 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → ¬ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
2524olcd 874 . 2 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (¬ 𝐹:𝐴𝐶 ∨ ¬ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
26 ianor 983 . . 3 (¬ (𝐹:𝐴𝐶 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ↔ (¬ 𝐹:𝐴𝐶 ∨ ¬ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
27 dff13 7275 . . 3 (𝐹:𝐴1-1𝐶 ↔ (𝐹:𝐴𝐶 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
2826, 27xchnxbir 333 . 2 𝐹:𝐴1-1𝐶 ↔ (¬ 𝐹:𝐴𝐶 ∨ ¬ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
2925, 28sylibr 234 1 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → ¬ 𝐹:𝐴1-1𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  {csn 4631  wf 6559  1-1wf1 6560  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fv 6571
This theorem is referenced by:  nf1oconst  7325
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