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Theorem nf1oconst 7326
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 7325 . . 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 6567 . . 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 2107  wne 2939  {csn 4625  wf 6556  1-1wf1 6557  ontowfo 6558  1-1-ontowf1o 6559
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-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-f1o 6567  df-fv 6568
This theorem is referenced by:  symgpssefmnd  19414
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