Theorem nf1const 7313

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} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → ¬ 𝐹:𝐴–1-1→𝐶)

Proof of Theorem nf1const

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1154	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝐴)
2		simp2 1155	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝐴)
3		fvconst 7168	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐵)
4	1, 3	sylan2 605	. . . . . . 7 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → (𝐹‘𝑋) = 𝐵)
5		fvconst 7168	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑌 ∈ 𝐴) → (𝐹‘𝑌) = 𝐵)
6	2, 5	sylan2 605	. . . . . . 7 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → (𝐹‘𝑌) = 𝐵)
7	4, 6	eqtr4d 2798	. . . . . 6 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → (𝐹‘𝑋) = (𝐹‘𝑌))
8		neneq 2961	. . . . . . . 8 ⊢ (𝑋 ≠ 𝑌 → ¬ 𝑋 = 𝑌)
9	8	3ad2ant3 1153	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌) → ¬ 𝑋 = 𝑌)
10	9	adantl 487	. . . . . 6 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → ¬ 𝑋 = 𝑌)
11	7, 10	jcnd 164	. . . . 5 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → ¬ ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))
12		fveqeq2 6895	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑋) = (𝐹‘𝑦)))
13		eqeq1 2764	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑦 ↔ 𝑋 = 𝑦))
14	12, 13	imbi12d 347	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ((𝐹‘𝑋) = (𝐹‘𝑦) → 𝑋 = 𝑦)))
15	14	notbid 321	. . . . . 6 ⊢ (𝑥 = 𝑋 → (¬ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ¬ ((𝐹‘𝑋) = (𝐹‘𝑦) → 𝑋 = 𝑦)))
16		fveq2 6886	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
17	16	eqeq2d 2771	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑋) = (𝐹‘𝑦) ↔ (𝐹‘𝑋) = (𝐹‘𝑌)))
18		eqeq2 2772	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑋 = 𝑦 ↔ 𝑋 = 𝑌))
19	17, 18	imbi12d 347	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (((𝐹‘𝑋) = (𝐹‘𝑦) → 𝑋 = 𝑦) ↔ ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))
20	19	notbid 321	. . . . . 6 ⊢ (𝑦 = 𝑌 → (¬ ((𝐹‘𝑋) = (𝐹‘𝑦) → 𝑋 = 𝑦) ↔ ¬ ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))
21	15, 20	rspc2ev 3588	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ ¬ ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
22	1, 2, 11, 21	syl2an23an 1450	. . . 4 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
23		rexnal2 3144	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
24	22, 23	sylib 221	. . 3 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
25	24	olcd 888	. 2 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → (¬ 𝐹:𝐴⟶𝐶 ∨ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
26		ianor 997	. . 3 ⊢ (¬ (𝐹:𝐴⟶𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) ↔ (¬ 𝐹:𝐴⟶𝐶 ∨ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
27		dff13 7259	. . 3 ⊢ (𝐹:𝐴–1-1→𝐶 ↔ (𝐹:𝐴⟶𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
28	26, 27	xchnxbir 336	. 2 ⊢ (¬ 𝐹:𝐴–1-1→𝐶 ↔ (¬ 𝐹:𝐴⟶𝐶 ∨ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
29	25, 28	sylibr 237	1 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → ¬ 𝐹:𝐴–1-1→𝐶)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {csn 4584  ⟶wf 6535  –1-1→wf1 6536  ‘cfv 6539

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fv 6547

This theorem is used by:  nf1oconst  7314