Theorem vonf1oonf1 35866

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–1-1-onto→On), 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–1-1-onto→On → 𝐻:𝐴–1-1→On)

Proof of Theorem vonf1oonf1

Step	Hyp	Ref	Expression
1		f1of1 6815	. . 3 ⊢ (𝐹:V–1-1-onto→On → 𝐹:V–1-1→On)
2		ssv 3955	. . 3 ⊢ 𝐴 ⊆ V
3		f1ssres 6779	. . 3 ⊢ ((𝐹:V–1-1→On ∧ 𝐴 ⊆ V) → (𝐹 ↾ 𝐴):𝐴–1-1→On)
4	1, 2, 3	sylancl 598	. 2 ⊢ (𝐹:V–1-1-onto→On → (𝐹 ↾ 𝐴):𝐴–1-1→On)
5		vonf1oonf1.1	. . 3 ⊢ 𝐻 = (𝐹 ↾ 𝐴)
6		f1eq1 6765	. . 3 ⊢ (𝐻 = (𝐹 ↾ 𝐴) → (𝐻:𝐴–1-1→On ↔ (𝐹 ↾ 𝐴):𝐴–1-1→On))
7	5, 6	ax-mp 5	. 2 ⊢ (𝐻:𝐴–1-1→On ↔ (𝐹 ↾ 𝐴):𝐴–1-1→On)
8	4, 7	sylibr 237	1 ⊢ (𝐹:V–1-1-onto→On → 𝐻:𝐴–1-1→On)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  Vcvv 3451   ⊆ wss 3899   ↾ cres 5653  Oncon0 6355  –1-1→wf1 6528  –1-1-onto→wf1o 6530

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

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 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-f1o 6538

This theorem is used by: (None)