Theorem vonf1oonf1 36077

Description: If 𝐹 is a bijection from the universe to the ordinals, then 𝐻 maps 𝐴 one-to-one into the ordinals. This is the ZFC version of (5 → 6) in https://tinyurl.com/hamkins-gblac. Note that in NBG set theory the antecedent would be something like ∀𝑋(¬ 𝑋 ∈ V → ∃𝐹𝐹:𝑋⟶1-1ontoOn), but since we cannot quantify over classes, we instead consider only the case 𝑋 = V which is sufficient for this proof. This theorem can also be viewed as (2 → 6). (Contributed by BTernaryTau, 10-Jun-2026.)

Hypothesis

Ref	Expression
vonf1oonf1.1	⊢ 𝐻 = (𝐹 ↾ 𝐴)

Assertion

Ref	Expression
vonf1oonf1	⊢ (𝐹:V⟶1-1ontoOn → 𝐻:𝐴⟶1-1On)

Proof of Theorem vonf1oonf1

Step	Hyp	Ref	Expression
1		f1of1 6851	. . 3 ⊢ (𝐹:V⟶1-1ontoOn → 𝐹:V⟶1-1On)
2		ssv 3937	. . 3 ⊢ 𝐴 ⊆ V
3		f1ssres 6815	. . 3 ⊢ ((𝐹:V⟶1-1On ∧ 𝐴 ⊆ V) → (𝐹 ↾ 𝐴):𝐴⟶1-1On)
4	1, 2, 3	sylancl 598	. 2 ⊢ (𝐹:V⟶1-1ontoOn → (𝐹 ↾ 𝐴):𝐴⟶1-1On)
5		vonf1oonf1.1	. . 3 ⊢ 𝐻 = (𝐹 ↾ 𝐴)
6		f1eq1 6801	. . 3 ⊢ (𝐻 = (𝐹 ↾ 𝐴) → (𝐻:𝐴⟶1-1On ↔ (𝐹 ↾ 𝐴):𝐴⟶1-1On))
7	5, 6	ax-mp 5	. 2 ⊢ (𝐻:𝐴⟶1-1On ↔ (𝐹 ↾ 𝐴):𝐴⟶1-1On)
8	4, 7	sylibr 237	1 ⊢ (𝐹:V⟶1-1ontoOn → 𝐻:𝐴⟶1-1On)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  Vcvv 3454   ⊆ wss 3900   ↾ cres 5658  Oncon0 6386  ⟶1-1wf1 6561  ⟶1-1ontowf1o 6563

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-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-f1o 6571

This theorem is used by: (None)