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Theorem nf1const 7032
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 1132 . . . . 5 ((𝑋𝐴𝑌𝐴𝑋𝑌) → 𝑋𝐴)
2 simp2 1133 . . . . 5 ((𝑋𝐴𝑌𝐴𝑋𝑌) → 𝑌𝐴)
3 fvconst 6898 . . . . . . . 8 ((𝐹:𝐴⟶{𝐵} ∧ 𝑋𝐴) → (𝐹𝑋) = 𝐵)
41, 3sylan2 594 . . . . . . 7 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (𝐹𝑋) = 𝐵)
5 fvconst 6898 . . . . . . . 8 ((𝐹:𝐴⟶{𝐵} ∧ 𝑌𝐴) → (𝐹𝑌) = 𝐵)
62, 5sylan2 594 . . . . . . 7 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (𝐹𝑌) = 𝐵)
74, 6eqtr4d 2858 . . . . . 6 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (𝐹𝑋) = (𝐹𝑌))
8 neneq 3012 . . . . . . . 8 (𝑋𝑌 → ¬ 𝑋 = 𝑌)
983ad2ant3 1131 . . . . . . 7 ((𝑋𝐴𝑌𝐴𝑋𝑌) → ¬ 𝑋 = 𝑌)
109adantl 484 . . . . . 6 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → ¬ 𝑋 = 𝑌)
117, 10jcnd 165 . . . . 5 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → ¬ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))
12 fveqeq2 6651 . . . . . . . 8 (𝑥 = 𝑋 → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑋) = (𝐹𝑦)))
13 eqeq1 2824 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 = 𝑦𝑋 = 𝑦))
1412, 13imbi12d 347 . . . . . . 7 (𝑥 = 𝑋 → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦)))
1514notbid 320 . . . . . 6 (𝑥 = 𝑋 → (¬ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ¬ ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦)))
16 fveq2 6642 . . . . . . . . 9 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
1716eqeq2d 2831 . . . . . . . 8 (𝑦 = 𝑌 → ((𝐹𝑋) = (𝐹𝑦) ↔ (𝐹𝑋) = (𝐹𝑌)))
18 eqeq2 2832 . . . . . . . 8 (𝑦 = 𝑌 → (𝑋 = 𝑦𝑋 = 𝑌))
1917, 18imbi12d 347 . . . . . . 7 (𝑦 = 𝑌 → (((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦) ↔ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))
2019notbid 320 . . . . . 6 (𝑦 = 𝑌 → (¬ ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦) ↔ ¬ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))
2115, 20rspc2ev 3611 . . . . 5 ((𝑋𝐴𝑌𝐴 ∧ ¬ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)) → ∃𝑥𝐴𝑦𝐴 ¬ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
221, 2, 11, 21syl2an23an 1419 . . . 4 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → ∃𝑥𝐴𝑦𝐴 ¬ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
23 rexnal2 3245 . . . 4 (∃𝑥𝐴𝑦𝐴 ¬ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ¬ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
2422, 23sylib 220 . . 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 6986 . . 3 (𝐹:𝐴1-1𝐶 ↔ (𝐹:𝐴𝐶 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
2826, 27xchnxbir 335 . 2 𝐹:𝐴1-1𝐶 ↔ (¬ 𝐹:𝐴𝐶 ∨ ¬ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
2925, 28sylibr 236 1 ((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → ¬ 𝐹:𝐴1-1𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  wo 843  w3a 1083   = wceq 1537  wcel 2114  wne 3006  wral 3125  wrex 3126  {csn 4539  wf 6323  1-1wf1 6324  cfv 6327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pr 5302
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-sbc 3749  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-opab 5101  df-id 5432  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fv 6335
This theorem is referenced by:  nf1oconst  7033
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