Theorem onelfvnef1 8460

Description: A sufficient condition for a function on ordinals to be one-to-one. (Contributed by NM, 9-Feb-1997.) Extract from tz7.48lem 8461 and generalize statement. (Revised by Matthew House, 6-Sep-2026.)

Assertion

Ref	Expression
onelfvnef1	⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → (𝐹‘𝑥) ≠ (𝐹‘𝑦))) → 𝐹:𝐴–1-1→𝐵)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐹,𝑦

Allowed substitution hints:   𝐵(𝑥, 𝑦)

Proof of Theorem onelfvnef1

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1154	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → (𝐹‘𝑥) ≠ (𝐹‘𝑦))) → 𝐹:𝐴⟶𝐵)
2		elequ12 2164	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑤))
3	2	ancoms 464	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑤))
4		fveq2 6893	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
5		fveq2 6893	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
6	4, 5	eqeqan12d 2777	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑤) = (𝐹‘𝑧)))
7	6	necon3bid 3002	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → ((𝐹‘𝑥) ≠ (𝐹‘𝑦) ↔ (𝐹‘𝑤) ≠ (𝐹‘𝑧)))
8	3, 7	imbi12d 347	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → ((𝑦 ∈ 𝑥 → (𝐹‘𝑥) ≠ (𝐹‘𝑦)) ↔ (𝑧 ∈ 𝑤 → (𝐹‘𝑤) ≠ (𝐹‘𝑧))))
9	8	rspc2gv 3588	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → (𝐹‘𝑥) ≠ (𝐹‘𝑦)) → (𝑧 ∈ 𝑤 → (𝐹‘𝑤) ≠ (𝐹‘𝑧))))
10	9	ancoms 464	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → (𝐹‘𝑥) ≠ (𝐹‘𝑦)) → (𝑧 ∈ 𝑤 → (𝐹‘𝑤) ≠ (𝐹‘𝑧))))
11	10	impcom 413	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → (𝐹‘𝑥) ≠ (𝐹‘𝑦)) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑧 ∈ 𝑤 → (𝐹‘𝑤) ≠ (𝐹‘𝑧)))
12		necom 3011	. . . . . . . 8 ⊢ ((𝐹‘𝑧) ≠ (𝐹‘𝑤) ↔ (𝐹‘𝑤) ≠ (𝐹‘𝑧))
13	11, 12	imbitrrdi 255	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → (𝐹‘𝑥) ≠ (𝐹‘𝑦)) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑧 ∈ 𝑤 → (𝐹‘𝑧) ≠ (𝐹‘𝑤)))
14		elequ12 2164	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) → (𝑦 ∈ 𝑥 ↔ 𝑤 ∈ 𝑧))
15	14	ancoms 464	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑦 ∈ 𝑥 ↔ 𝑤 ∈ 𝑧))
16		fveq2 6893	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
17		fveq2 6893	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
18	16, 17	eqeqan12d 2777	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑧) = (𝐹‘𝑤)))
19	18	necon3bid 3002	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝐹‘𝑥) ≠ (𝐹‘𝑦) ↔ (𝐹‘𝑧) ≠ (𝐹‘𝑤)))
20	15, 19	imbi12d 347	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑦 ∈ 𝑥 → (𝐹‘𝑥) ≠ (𝐹‘𝑦)) ↔ (𝑤 ∈ 𝑧 → (𝐹‘𝑧) ≠ (𝐹‘𝑤))))
21	20	rspc2gv 3588	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → (𝐹‘𝑥) ≠ (𝐹‘𝑦)) → (𝑤 ∈ 𝑧 → (𝐹‘𝑧) ≠ (𝐹‘𝑤))))
22	21	impcom 413	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → (𝐹‘𝑥) ≠ (𝐹‘𝑦)) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑤 ∈ 𝑧 → (𝐹‘𝑧) ≠ (𝐹‘𝑤)))
23	13, 22	jaod 873	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → (𝐹‘𝑥) ≠ (𝐹‘𝑦)) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑧 ∈ 𝑤 ∨ 𝑤 ∈ 𝑧) → (𝐹‘𝑧) ≠ (𝐹‘𝑤)))
24	23	necon2bd 2974	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → (𝐹‘𝑥) ≠ (𝐹‘𝑦)) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → ¬ (𝑧 ∈ 𝑤 ∨ 𝑤 ∈ 𝑧)))
25	24	3ad2antl3 1206	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → (𝐹‘𝑥) ≠ (𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → ¬ (𝑧 ∈ 𝑤 ∨ 𝑤 ∈ 𝑧)))
26		ssel2 3907	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ On)
27		ssel2 3907	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ On)
28		eloni 6378	. . . . . . . 8 ⊢ (𝑧 ∈ On → Ord 𝑧)
29		eloni 6378	. . . . . . . 8 ⊢ (𝑤 ∈ On → Ord 𝑤)
30		ordtri3 6405	. . . . . . . 8 ⊢ ((Ord 𝑧 ∧ Ord 𝑤) → (𝑧 = 𝑤 ↔ ¬ (𝑧 ∈ 𝑤 ∨ 𝑤 ∈ 𝑧)))
31	28, 29, 30	syl2an 608	. . . . . . 7 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧 = 𝑤 ↔ ¬ (𝑧 ∈ 𝑤 ∨ 𝑤 ∈ 𝑧)))
32	26, 27, 31	syl2an 608	. . . . . 6 ⊢ (((𝐴 ⊆ On ∧ 𝑧 ∈ 𝐴) ∧ (𝐴 ⊆ On ∧ 𝑤 ∈ 𝐴)) → (𝑧 = 𝑤 ↔ ¬ (𝑧 ∈ 𝑤 ∨ 𝑤 ∈ 𝑧)))
33	32	anandis 691	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑧 = 𝑤 ↔ ¬ (𝑧 ∈ 𝑤 ∨ 𝑤 ∈ 𝑧)))
34	33	3ad2antl2 1205	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → (𝐹‘𝑥) ≠ (𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑧 = 𝑤 ↔ ¬ (𝑧 ∈ 𝑤 ∨ 𝑤 ∈ 𝑧)))
35	25, 34	sylibrd 262	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → (𝐹‘𝑥) ≠ (𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
36	35	ralrimivva 3208	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → (𝐹‘𝑥) ≠ (𝐹‘𝑦))) → ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
37		dff13 7265	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)))
38	1, 36, 37	sylanbrc 595	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → (𝐹‘𝑥) ≠ (𝐹‘𝑦))) → 𝐹:𝐴–1-1→𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146   ≠ wne 2958  ∀wral 3079   ⊆ wss 3899  Ord word 6367  Oncon0 6368  ⟶wf 6542  –1-1→wf1 6543  ‘cfv 6546

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-ord 6371  df-on 6372  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fv 6554

This theorem is used by:  tz7.48lem  8461  mh-inf3f1  37456