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Theorem nf1const 7161
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 1134 . . . . 5 ((𝑋𝐴𝑌𝐴𝑋𝑌) → 𝑋𝐴)
2 simp2 1135 . . . . 5 ((𝑋𝐴𝑌𝐴𝑋𝑌) → 𝑌𝐴)
3 fvconst 7023 . . . . . . . 8 ((𝐹:𝐴⟶{𝐵} ∧ 𝑋𝐴) → (𝐹𝑋) = 𝐵)
41, 3sylan2 592 . . . . . . 7 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (𝐹𝑋) = 𝐵)
5 fvconst 7023 . . . . . . . 8 ((𝐹:𝐴⟶{𝐵} ∧ 𝑌𝐴) → (𝐹𝑌) = 𝐵)
62, 5sylan2 592 . . . . . . 7 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (𝐹𝑌) = 𝐵)
74, 6eqtr4d 2780 . . . . . 6 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (𝐹𝑋) = (𝐹𝑌))
8 neneq 2947 . . . . . . . 8 (𝑋𝑌 → ¬ 𝑋 = 𝑌)
983ad2ant3 1133 . . . . . . 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 6770 . . . . . . . 8 (𝑥 = 𝑋 → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑋) = (𝐹𝑦)))
13 eqeq1 2741 . . . . . . . 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 6761 . . . . . . . . 9 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
1716eqeq2d 2748 . . . . . . . 8 (𝑦 = 𝑌 → ((𝐹𝑋) = (𝐹𝑦) ↔ (𝐹𝑋) = (𝐹𝑌)))
18 eqeq2 2749 . . . . . . . 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 3569 . . . . 5 ((𝑋𝐴𝑌𝐴 ∧ ¬ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)) → ∃𝑥𝐴𝑦𝐴 ¬ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
221, 2, 11, 21syl2an23an 1421 . . . 4 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → ∃𝑥𝐴𝑦𝐴 ¬ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
23 rexnal2 3185 . . . 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 870 . 2 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (¬ 𝐹:𝐴𝐶 ∨ ¬ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
26 ianor 978 . . 3 (¬ (𝐹:𝐴𝐶 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ↔ (¬ 𝐹:𝐴𝐶 ∨ ¬ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
27 dff13 7114 . . 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 395  wo 843  w3a 1085   = wceq 1539  wcel 2107  wne 2941  wral 3062  wrex 3063  {csn 4563  wf 6419  1-1wf1 6420  cfv 6423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3429  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-br 5076  df-opab 5138  df-id 5485  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-iota 6381  df-fun 6425  df-fn 6426  df-f 6427  df-f1 6428  df-fv 6431
This theorem is referenced by:  nf1oconst  7162
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