MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nf1oconst Structured version   Visualization version   GIF version

Theorem nf1oconst 7320
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 7319 . . 3 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → ¬ 𝐹:𝐴1-1𝐶)
21orcd 871 . 2 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (¬ 𝐹:𝐴1-1𝐶 ∨ ¬ 𝐹:𝐴onto𝐶))
3 ianor 979 . . 3 (¬ (𝐹:𝐴1-1𝐶𝐹:𝐴onto𝐶) ↔ (¬ 𝐹:𝐴1-1𝐶 ∨ ¬ 𝐹:𝐴onto𝐶))
4 df-f1o 6560 . . 3 (𝐹:𝐴1-1-onto𝐶 ↔ (𝐹:𝐴1-1𝐶𝐹:𝐴onto𝐶))
53, 4xchnxbir 332 . 2 𝐹:𝐴1-1-onto𝐶 ↔ (¬ 𝐹:𝐴1-1𝐶 ∨ ¬ 𝐹:𝐴onto𝐶))
62, 5sylibr 233 1 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → ¬ 𝐹:𝐴1-1-onto𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wo 845  w3a 1084  wcel 2098  wne 2937  {csn 4632  wf 6549  1-1wf1 6550  ontowfo 6551  1-1-ontowf1o 6552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-f1o 6560  df-fv 6561
This theorem is referenced by:  symgpssefmnd  19364
  Copyright terms: Public domain W3C validator