Theorem nf1const 7338

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⟶1-1𝐶)

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 7194	. . . . . . . 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 7194	. . . . . . . 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 2801	. . . . . 6 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → (𝐹‘𝑋) = (𝐹‘𝑌))
8		neneq 2965	. . . . . . . 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 6920	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑋) = (𝐹‘𝑦)))
13		eqeq1 2767	. . . . . . . 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 6911	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
17	16	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑋) = (𝐹‘𝑦) ↔ (𝐹‘𝑋) = (𝐹‘𝑌)))
18		eqeq2 2775	. . . . . . . 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 3592	. . . . 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 1451	. . . 4 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
23		rexnal2 3148	. . . 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 7284	. . 3 ⊢ (𝐹:𝐴⟶1-1𝐶 ↔ (𝐹:𝐴⟶𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
28	26, 27	xchnxbir 336	. 2 ⊢ (¬ 𝐹:𝐴⟶1-1𝐶 ↔ (¬ 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⟶1-1𝐶)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  {csn 4588  ⟶wf 6557  ⟶1-1wf1 6558  ‘cfv 6561

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fv 6569

This theorem is used by:  nf1oconst  7339