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Theorem nf1const 7304
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 1136 . . . . 5 ((𝑋𝐴𝑌𝐴𝑋𝑌) → 𝑋𝐴)
2 simp2 1137 . . . . 5 ((𝑋𝐴𝑌𝐴𝑋𝑌) → 𝑌𝐴)
3 fvconst 7164 . . . . . . . 8 ((𝐹:𝐴⟶{𝐵} ∧ 𝑋𝐴) → (𝐹𝑋) = 𝐵)
41, 3sylan2 593 . . . . . . 7 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (𝐹𝑋) = 𝐵)
5 fvconst 7164 . . . . . . . 8 ((𝐹:𝐴⟶{𝐵} ∧ 𝑌𝐴) → (𝐹𝑌) = 𝐵)
62, 5sylan2 593 . . . . . . 7 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (𝐹𝑌) = 𝐵)
74, 6eqtr4d 2775 . . . . . 6 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (𝐹𝑋) = (𝐹𝑌))
8 neneq 2946 . . . . . . . 8 (𝑋𝑌 → ¬ 𝑋 = 𝑌)
983ad2ant3 1135 . . . . . . 7 ((𝑋𝐴𝑌𝐴𝑋𝑌) → ¬ 𝑋 = 𝑌)
109adantl 482 . . . . . 6 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → ¬ 𝑋 = 𝑌)
117, 10jcnd 163 . . . . 5 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → ¬ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))
12 fveqeq2 6900 . . . . . . . 8 (𝑥 = 𝑋 → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑋) = (𝐹𝑦)))
13 eqeq1 2736 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 = 𝑦𝑋 = 𝑦))
1412, 13imbi12d 344 . . . . . . 7 (𝑥 = 𝑋 → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦)))
1514notbid 317 . . . . . 6 (𝑥 = 𝑋 → (¬ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ¬ ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦)))
16 fveq2 6891 . . . . . . . . 9 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
1716eqeq2d 2743 . . . . . . . 8 (𝑦 = 𝑌 → ((𝐹𝑋) = (𝐹𝑦) ↔ (𝐹𝑋) = (𝐹𝑌)))
18 eqeq2 2744 . . . . . . . 8 (𝑦 = 𝑌 → (𝑋 = 𝑦𝑋 = 𝑌))
1917, 18imbi12d 344 . . . . . . 7 (𝑦 = 𝑌 → (((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦) ↔ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))
2019notbid 317 . . . . . 6 (𝑦 = 𝑌 → (¬ ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦) ↔ ¬ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))
2115, 20rspc2ev 3624 . . . . 5 ((𝑋𝐴𝑌𝐴 ∧ ¬ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)) → ∃𝑥𝐴𝑦𝐴 ¬ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
221, 2, 11, 21syl2an23an 1423 . . . 4 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → ∃𝑥𝐴𝑦𝐴 ¬ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
23 rexnal2 3135 . . . 4 (∃𝑥𝐴𝑦𝐴 ¬ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ¬ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
2422, 23sylib 217 . . 3 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → ¬ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
2524olcd 872 . 2 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (¬ 𝐹:𝐴𝐶 ∨ ¬ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
26 ianor 980 . . 3 (¬ (𝐹:𝐴𝐶 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ↔ (¬ 𝐹:𝐴𝐶 ∨ ¬ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
27 dff13 7256 . . 3 (𝐹:𝐴1-1𝐶 ↔ (𝐹:𝐴𝐶 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
2826, 27xchnxbir 332 . 2 𝐹:𝐴1-1𝐶 ↔ (¬ 𝐹:𝐴𝐶 ∨ ¬ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
2925, 28sylibr 233 1 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → ¬ 𝐹:𝐴1-1𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2940  wral 3061  wrex 3070  {csn 4628  wf 6539  1-1wf1 6540  cfv 6543
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fv 6551
This theorem is referenced by:  nf1oconst  7305
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