Theorem vonf1owev 36037

Description: If 𝐹 is a bijection from the universe to the ordinals, then 𝑅 well-orders the universe. This is the ZFC version of (2 → 3) in https://tinyurl.com/hamkins-gblac. (Contributed by BTernaryTau, 6-Dec-2025.) (Proof shortened by BTernaryTau, 11-Jun-2026.)

Hypothesis

Ref	Expression
vonf1owev.1	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)}

Assertion

Ref	Expression
vonf1owev	⊢ (𝐹:V⟶1-1ontoOn → 𝑅 We V)

Distinct variable group:   𝑥,𝐹,𝑦

Allowed substitution hints:   𝑅(𝑥, 𝑦)

Proof of Theorem vonf1owev

Step	Hyp	Ref	Expression
1		f1of1 6845	. 2 ⊢ (𝐹:V⟶1-1ontoOn → 𝐹:V⟶1-1On)
2		vonf1owev.1	. . 3 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)}
3	2	vonf1wev 36036	. 2 ⊢ (𝐹:V⟶1-1On → 𝑅 We V)
4	1, 3	syl 18	1 ⊢ (𝐹:V⟶1-1ontoOn → 𝑅 We V)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454  {copab 5170   We wwe 5607  Oncon0 6380  ⟶1-1wf1 6555  ⟶1-1ontowf1o 6557  ‘cfv 6558

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-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5171  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6383  df-on 6384  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-f1o 6565  df-fv 6566

This theorem is used by:  vonf1osev  36040