Theorem nf1const 7208

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 1136	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝐴)
2		simp2 1137	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝐴)
3		fvconst 7068	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐵)
4	1, 3	sylan2 594	. . . . . . 7 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → (𝐹‘𝑋) = 𝐵)
5		fvconst 7068	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑌 ∈ 𝐴) → (𝐹‘𝑌) = 𝐵)
6	2, 5	sylan2 594	. . . . . . 7 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → (𝐹‘𝑌) = 𝐵)
7	4, 6	eqtr4d 2779	. . . . . 6 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → (𝐹‘𝑋) = (𝐹‘𝑌))
8		neneq 2947	. . . . . . . 8 ⊢ (𝑋 ≠ 𝑌 → ¬ 𝑋 = 𝑌)
9	8	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌) → ¬ 𝑋 = 𝑌)
10	9	adantl 483	. . . . . 6 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → ¬ 𝑋 = 𝑌)
11	7, 10	jcnd 163	. . . . 5 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → ¬ ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))
12		fveqeq2 6813	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑋) = (𝐹‘𝑦)))
13		eqeq1 2740	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑦 ↔ 𝑋 = 𝑦))
14	12, 13	imbi12d 345	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ((𝐹‘𝑋) = (𝐹‘𝑦) → 𝑋 = 𝑦)))
15	14	notbid 318	. . . . . 6 ⊢ (𝑥 = 𝑋 → (¬ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ¬ ((𝐹‘𝑋) = (𝐹‘𝑦) → 𝑋 = 𝑦)))
16		fveq2 6804	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
17	16	eqeq2d 2747	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑋) = (𝐹‘𝑦) ↔ (𝐹‘𝑋) = (𝐹‘𝑌)))
18		eqeq2 2748	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑋 = 𝑦 ↔ 𝑋 = 𝑌))
19	17, 18	imbi12d 345	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (((𝐹‘𝑋) = (𝐹‘𝑦) → 𝑋 = 𝑦) ↔ ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))
20	19	notbid 318	. . . . . 6 ⊢ (𝑦 = 𝑌 → (¬ ((𝐹‘𝑋) = (𝐹‘𝑦) → 𝑋 = 𝑦) ↔ ¬ ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))
21	15, 20	rspc2ev 3577	. . . . 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 1423	. . . 4 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
23		rexnal2 3129	. . . 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 217	. . 3 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
25	24	olcd 872	. 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 980	. . 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 7160	. . 3 ⊢ (𝐹:𝐴–1-1→𝐶 ↔ (𝐹:𝐴⟶𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
28	26, 27	xchnxbir 333	. 2 ⊢ (¬ 𝐹:𝐴–1-1→𝐶 ↔ (¬ 𝐹:𝐴⟶𝐶 ∨ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
29	25, 28	sylibr 233	1 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → ¬ 𝐹:𝐴–1-1→𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   ∨ wo 845   ∧ w3a 1087   = wceq 1539   ∈ wcel 2104   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  {csn 4565  ⟶wf 6454  –1-1→wf1 6455  ‘cfv 6458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fv 6466

This theorem is referenced by:  nf1oconst  7209