Theorem vonf1oonf1 35576

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 6821	. . 3 ⊢ (𝐹:V–1-1-onto→On → 𝐹:V–1-1→On)
2		ssv 3962	. . 3 ⊢ 𝐴 ⊆ V
3		f1ssres 6785	. . 3 ⊢ ((𝐹:V–1-1→On ∧ 𝐴 ⊆ V) → (𝐹 ↾ 𝐴):𝐴–1-1→On)
4	1, 2, 3	sylancl 597	. 2 ⊢ (𝐹:V–1-1-onto→On → (𝐹 ↾ 𝐴):𝐴–1-1→On)
5		vonf1oonf1.1	. . 3 ⊢ 𝐻 = (𝐹 ↾ 𝐴)
6		f1eq1 6771	. . 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)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570  Vcvv 3455   ⊆ wss 3906   ↾ cres 5665  Oncon0 6362  –1-1→wf1 6535  –1-1-onto→wf1o 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-f1o 6545

This theorem is referenced by: (None)