Theorem fdcf1 7306

Description: It is decidable whether a function from a finite set into another finite set is one-to-one. (Contributed by Jim Kingdon, 13-Jul-2026.)

Assertion

Ref	Expression
fdcf1	⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴⟶𝐵) → DECID 𝐹:𝐴–1-1→𝐵)

Proof of Theorem fdcf1

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

Step	Hyp	Ref	Expression
1		simp3 1030	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴⟶𝐵)
2	1	orcd 745	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴⟶𝐵) → (𝐹:𝐴⟶𝐵 ∨ ¬ 𝐹:𝐴⟶𝐵))
3		df-dc 847	. . . 4 ⊢ (DECID 𝐹:𝐴⟶𝐵 ↔ (𝐹:𝐴⟶𝐵 ∨ ¬ 𝐹:𝐴⟶𝐵))
4	2, 3	sylibr 134	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴⟶𝐵) → DECID 𝐹:𝐴⟶𝐵)
5		simp1 1028	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴⟶𝐵) → 𝐴 ∈ Fin)
6	5	adantr 276	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ Fin)
7		simpll2 1068	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝐵 ∈ Fin)
8		simpll3 1069	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝐹:𝐴⟶𝐵)
9		simplr 533	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴)
10	8, 9	ffvelcdmd 5835	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
11		simpr 110	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
12	8, 11	ffvelcdmd 5835	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ 𝐵)
13		fidceq 7161	. . . . . . . . 9 ⊢ ((𝐵 ∈ Fin ∧ (𝐹‘𝑥) ∈ 𝐵 ∧ (𝐹‘𝑦) ∈ 𝐵) → DECID (𝐹‘𝑥) = (𝐹‘𝑦))
14	7, 10, 12, 13	syl3anc 1278	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → DECID (𝐹‘𝑥) = (𝐹‘𝑦))
15	6	adantr 276	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝐴 ∈ Fin)
16		fidceq 7161	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → DECID 𝑥 = 𝑦)
17	15, 9, 11, 16	syl3anc 1278	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → DECID 𝑥 = 𝑦)
18		dcim 853	. . . . . . . 8 ⊢ (DECID (𝐹‘𝑥) = (𝐹‘𝑦) → (DECID 𝑥 = 𝑦 → DECID ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
19	14, 17, 18	sylc 62	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → DECID ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
20	19	ralrimiva 2623	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 DECID ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
21		dcfi 7305	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑦 ∈ 𝐴 DECID ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) → DECID ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
22	6, 20, 21	syl2anc 415	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → DECID ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
23	22	ralrimiva 2623	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴⟶𝐵) → ∀𝑥 ∈ 𝐴 DECID ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
24		dcfi 7305	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) → DECID ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
25	5, 23, 24	syl2anc 415	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴⟶𝐵) → DECID ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
26	4, 25	dcand 945	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴⟶𝐵) → DECID (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
27		dff13 5964	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
28	27	dcbii 852	. 2 ⊢ (DECID 𝐹:𝐴–1-1→𝐵 ↔ DECID (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
29	26, 28	sylibr 134	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴⟶𝐵) → DECID 𝐹:𝐴–1-1→𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720  DECID wdc 846   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ⟶wf 5368  –1-1→wf1 5369  ‘cfv 5372  Fincfn 7012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-er 6797  df-en 7013  df-fin 7015

This theorem is referenced by:  f1setfi  7307