Theorem vonf1oonf1 35469

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 6809	. . 3 ⊢ (𝐹:V–1-1-onto→On → 𝐹:V–1-1→On)
2		ssv 3963	. . 3 ⊢ 𝐴 ⊆ V
3		f1ssres 6773	. . 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 6759	. . 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 1563  Vcvv 3457   ⊆ wss 3907   ↾ cres 5654  Oncon0 6350  –1-1→wf1 6522  –1-1-onto→wf1o 6524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-f1o 6532

This theorem is referenced by: (None)