Theorem tz7.48lem 8450

Description: A way of showing an ordinal function is one-to-one. (Contributed by NM, 9-Feb-1997.) Extract onelfvnef1 8449. (Proof shortened by Matthew House, 6-Sep-2026.)

Hypothesis

Ref	Expression
tz7.48.1	⊢ 𝐹 Fn On

Assertion

Ref	Expression
tz7.48lem	⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → Fun ◡(𝐹 ↾ 𝐴))

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

Proof of Theorem tz7.48lem

Step	Hyp	Ref	Expression
1		tz7.48.1	. . . . . 6 ⊢ 𝐹 Fn On
2		dffn2 6711	. . . . . 6 ⊢ (𝐹 Fn On ↔ 𝐹:On⟶V)
3	1, 2	mpbi 233	. . . . 5 ⊢ 𝐹:On⟶V
4		fssres 6748	. . . . 5 ⊢ ((𝐹:On⟶V ∧ 𝐴 ⊆ On) → (𝐹 ↾ 𝐴):𝐴⟶V)
5	3, 4	mpan 703	. . . 4 ⊢ (𝐴 ⊆ On → (𝐹 ↾ 𝐴):𝐴⟶V)
6		fvres 6904	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
7		fvres 6904	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
8	6, 7	eqeqan12d 2774	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
9	8	necon3abid 2991	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (((𝐹 ↾ 𝐴)‘𝑥) ≠ ((𝐹 ↾ 𝐴)‘𝑦) ↔ ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
10	9	imbi2d 343	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑦 ∈ 𝑥 → ((𝐹 ↾ 𝐴)‘𝑥) ≠ ((𝐹 ↾ 𝐴)‘𝑦)) ↔ (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
11	10	exbiri 823	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐴 → ((𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝑦 ∈ 𝑥 → ((𝐹 ↾ 𝐴)‘𝑥) ≠ ((𝐹 ↾ 𝐴)‘𝑦)))))
12	11	com23 87	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ((𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝑦 ∈ 𝐴 → (𝑦 ∈ 𝑥 → ((𝐹 ↾ 𝐴)‘𝑥) ≠ ((𝐹 ↾ 𝐴)‘𝑦)))))
13	12	ralimdv2 3171	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦) → ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → ((𝐹 ↾ 𝐴)‘𝑥) ≠ ((𝐹 ↾ 𝐴)‘𝑦))))
14	13	ralimia 3096	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → ((𝐹 ↾ 𝐴)‘𝑥) ≠ ((𝐹 ↾ 𝐴)‘𝑦)))
15		onelfvnef1 8449	. . . . 5 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶V ∧ 𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → ((𝐹 ↾ 𝐴)‘𝑥) ≠ ((𝐹 ↾ 𝐴)‘𝑦))) → (𝐹 ↾ 𝐴):𝐴–1-1→V)
16	14, 15	syl3an3 1183	. . . 4 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶V ∧ 𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐹 ↾ 𝐴):𝐴–1-1→V)
17	5, 16	syl3an1 1181	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐹 ↾ 𝐴):𝐴–1-1→V)
18	17	3anidm12 1446	. 2 ⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐹 ↾ 𝐴):𝐴–1-1→V)
19		df-f1 6543	. . 3 ⊢ ((𝐹 ↾ 𝐴):𝐴–1-1→V ↔ ((𝐹 ↾ 𝐴):𝐴⟶V ∧ Fun ◡(𝐹 ↾ 𝐴)))
20	19	simprbi 503	. 2 ⊢ ((𝐹 ↾ 𝐴):𝐴–1-1→V → Fun ◡(𝐹 ↾ 𝐴))
21	18, 20	syl 18	1 ⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → Fun ◡(𝐹 ↾ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  Vcvv 3450   ⊆ wss 3895  ^◡ccnv 5649   ↾ cres 5652  Oncon0 6362  Fun wfun 6532   Fn wfn 6533  ⟶wf 6534  –1-1→wf1 6535  ‘cfv 6538

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-3or 1104  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-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ord 6365  df-on 6366  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fv 6546

This theorem is used by:  tz7.48-2  8452  tz7.49  8455  zorn2lem4  10577