Theorem tz7.48lem 8475

Description: A way of showing an ordinal function is one-to-one. (Contributed by NM, 9-Feb-1997.) Extract onelfvnef1 8474. (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 6730	. . . . . 6 ⊢ (𝐹 Fn On ↔ 𝐹:On⟶V)
3	1, 2	mpbi 233	. . . . 5 ⊢ 𝐹:On⟶V
4		fssres 6767	. . . . 5 ⊢ ((𝐹:On⟶V ∧ 𝐴 ⊆ On) → (𝐹 ↾ 𝐴):𝐴⟶V)
5	3, 4	mpan 703	. . . 4 ⊢ (𝐴 ⊆ On → (𝐹 ↾ 𝐴):𝐴⟶V)
6		fvres 6924	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
7		fvres 6924	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
8	6, 7	eqeqan12d 2777	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
9	8	necon3abid 2995	. . . . . . . . . 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 3175	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦) → ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → ((𝐹 ↾ 𝐴)‘𝑥) ≠ ((𝐹 ↾ 𝐴)‘𝑦))))
14	13	ralimia 3100	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → ((𝐹 ↾ 𝐴)‘𝑥) ≠ ((𝐹 ↾ 𝐴)‘𝑦)))
15		onelfvnef1 8474	. . . . 5 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶V ∧ 𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → ((𝐹 ↾ 𝐴)‘𝑥) ≠ ((𝐹 ↾ 𝐴)‘𝑦))) → (𝐹 ↾ 𝐴):𝐴⟶1-1V)
16	14, 15	syl3an3 1183	. . . 4 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶V ∧ 𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐹 ↾ 𝐴):𝐴⟶1-1V)
17	5, 16	syl3an1 1181	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐹 ↾ 𝐴):𝐴⟶1-1V)
18	17	3anidm12 1447	. 2 ⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐹 ↾ 𝐴):𝐴⟶1-1V)
19		df-f1 6560	. . 3 ⊢ ((𝐹 ↾ 𝐴):𝐴⟶1-1V ↔ ((𝐹 ↾ 𝐴):𝐴⟶V ∧ Fun ◡(𝐹 ↾ 𝐴)))
20	19	simprbi 503	. 2 ⊢ ((𝐹 ↾ 𝐴):𝐴⟶1-1V → 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 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  Vcvv 3454   ⊆ wss 3900  ^◡ccnv 5654   ↾ cres 5657  Oncon0 6377  Fun wfun 6549   Fn wfn 6550  ⟶wf 6551  ⟶1-1wf1 6552  ‘cfv 6555

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 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  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-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ord 6380  df-on 6381  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fv 6563

This theorem is used by:  tz7.48-2  8477  tz7.49  8480  zorn2lem4  10601