Theorem vonf1oonf1 35421

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 6801	. . 3 ⊢ (𝐹:V–1-1-onto→On → 𝐹:V–1-1→On)
2		ssv 3960	. . 3 ⊢ 𝐴 ⊆ V
3		f1ssres 6765	. . 3 ⊢ ((𝐹:V–1-1→On ∧ 𝐴 ⊆ V) → (𝐹 ↾ 𝐴):𝐴–1-1→On)
4	1, 2, 3	sylancl 595	. 2 ⊢ (𝐹:V–1-1-onto→On → (𝐹 ↾ 𝐴):𝐴–1-1→On)
5		vonf1oonf1.1	. . 3 ⊢ 𝐻 = (𝐹 ↾ 𝐴)
6		f1eq1 6751	. . 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 236	1 ⊢ (𝐹:V–1-1-onto→On → 𝐻:𝐴–1-1→On)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1559  Vcvv 3453   ⊆ wss 3904   ↾ cres 5647  Oncon0 6342  –1-1→wf1 6514  –1-1-onto→wf1o 6516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-f1o 6524

This theorem is referenced by: (None)