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Theorem nf1oconst 7325
Description: A constant function from at least two elements is not bijective. (Contributed by AV, 30-Mar-2024.)
Assertion
Ref Expression
nf1oconst ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → ¬ 𝐹:𝐴1-1-onto𝐶)

Proof of Theorem nf1oconst
StepHypRef Expression
1 nf1const 7324 . . 3 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → ¬ 𝐹:𝐴1-1𝐶)
21orcd 873 . 2 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (¬ 𝐹:𝐴1-1𝐶 ∨ ¬ 𝐹:𝐴onto𝐶))
3 ianor 983 . . 3 (¬ (𝐹:𝐴1-1𝐶𝐹:𝐴onto𝐶) ↔ (¬ 𝐹:𝐴1-1𝐶 ∨ ¬ 𝐹:𝐴onto𝐶))
4 df-f1o 6570 . . 3 (𝐹:𝐴1-1-onto𝐶 ↔ (𝐹:𝐴1-1𝐶𝐹:𝐴onto𝐶))
53, 4xchnxbir 333 . 2 𝐹:𝐴1-1-onto𝐶 ↔ (¬ 𝐹:𝐴1-1𝐶 ∨ ¬ 𝐹:𝐴onto𝐶))
62, 5sylibr 234 1 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → ¬ 𝐹:𝐴1-1-onto𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847  w3a 1086  wcel 2106  wne 2938  {csn 4631  wf 6559  1-1wf1 6560  ontowfo 6561  1-1-ontowf1o 6562
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-f1o 6570  df-fv 6571
This theorem is referenced by:  symgpssefmnd  19428
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